EQUATIONS FOR THE CALCULATION OF
CHROMATOGRAPHIC FIGURES OF MERIT
By
JOE PRESTON FOLEY
A DISSERTATION PRESENTED TO THE GRADUATE SCHOOI
OF THE UNIVERSITY OF FLORIDA IN
PARTIAL FULFILLMENT OF THE REQUIREMENTS
FOR THE DEGREE OF DOCTOR OF PHILOSOPHY
UNIVERSITY OF FLORIDA
1983
To Mom and Dad, for their love, support, and encouragement;
and to my sister Barbara, for leading the way.
ACKNOWLEDGMENTS
First and foremost, I would like to thank my research director,
John G. Dorsey, not only for helping me with various research projects,
but for being a special friend— for always finding the time to listen
and for always doing the "extra things."
Second, I want to express my gratitude to Thomas J. Buckley and
Sharon G. Lias of the National Bureau of Standards for their help and
the use of their facilities in preparing this document.
Finally, I want to thank all the friends I made in Gainesville for
making my graduate education at the University of Florida the happiest
and most satisfying time of my life.
111
TABLE OF CONTENTS
CHAPTER PAGE
ACKNOWLEDGMENTS iii
ABSTRACT v i
1 INTRODUCTION 1
Overview „ 1
Chromatographic Peak Characterization 1
Limit of Detection 8
2 GENERATION OF THE EXPONENTIALLY MODIFIED GAUSSIAN (EMG)
FUNCTION AND RELATED DATA 9
Introduction 9
EMG Evaluation 9
Background 9
Evaluation of the Integral Term 10
Obtaining Universal EMG Data 16
Background 16
Measurement of the Pertinent Peak Parameters . „ 17
Comparison of Universal EMG Data 22
Conclusion „ 24
3 EQUATIONS FOR THE CALCULATION OF CHROMATOGRAPHIC
FIGURES OF MERIT FOR IDEAL AND SKEWED PEAKS 25
Introduction 25
Derivations 25
Experimental , 28
Apparatus , 28
Procedure 28
Results 30
Recommended CFOM Equations , 30
Other CFOM Equations 33
Discussion _ 39
Detailed Discussion of Precision 39
Why Measure At 10$ Peak Height? 41
General Aspects , 46
Conclusion , 55
IV
4 CLARIFICATION OF THE LIMIT OF DETECTION IN CHROMATOGRAPHY 56
Introduction . 5b
Identifying Current Problems 56
Literature Survey Results 56
Numerical Example 58
Solving the Problems „ 60
Eliminating Mistaken Identities 60
Choosing a Model 63
Using the Correct Units 66
Converting to Chromatographic Reference Conditions 70
The Numerical Example  Revisited 79
Conclusion 81
5 SUGGESTIONS FOR FUTURE WORK 82
APPENDICES
A DISCUSSION AND LISTING OF THE BASIC PROGRAM, EMGU 84
B UNIVERSAL EMG DATA AT a = 0.05, 0.10, 0.30, and 0.50 87
C DERIVATION OF V inj>max 95
REFERENCES 96
BIOGRAPHICAL SKETCH 100
Abstract of Dissertation Presented to the Graduate School
of the University of Florida in Partial Fulfillment of the
Requirements for the Degree of Doctor of Philosophy
EQUATIONS FOR THE CALCULATION OF
CHROMATOGRAPHIC FIGURES OF MERIT
By
JOE PRESTON FOLEY
December 1 983
Chairman: Dr. John G. Dorsey
Major Department: Chemistry
The measurement and interpretation of several chromatographic
concepts and parameters, hereafter referred to as chromatographic
figures of merit (CFOMs) , are improved via equations and concepts
developed in this work.
The previous uses of the exponentially modified Gaussian (EMG)
model in chromatography are briefly reviewed. A method for evaluating
the EMG function and a set of algorithms for obtaining universal data
are presented and are shown to be simpler and easier to use than those
previously reported. The corresponding BASIC program, EMGU, is also
briefly discussed.
By use of the exponentially modified Gaussian (EMG) as the skewed
peak model, empirical equations based solely on the graphically
measurable retention time, t fi , peak width at 10% peak height, W Q ,, and
vi
the empirical asymmetry factor, B/A, have been developed for the
accurate and precise calculation of CFOMs characterizing both ideal
(Gaussian) and skewed peaks. These CFOMs include the observed
efficiency (number of theoretical plates), N ; the maximum efficiency
sys
attainable if all asymmetry is eliminated, N„ v ; the EMG peak
parameters, t^, a,, and t; the first through fourth statistical moments;
the peak skew and peak excess, Yg and Y„; and two new CFOMs — the
relative system efficiency, RSE, and the relative plate loss, RPL.
Equations for the number of theoretical plates and the variance (second
central moment) are accurate to within ±1.5% for 1.00 <_ B/A £ 2.76.
width and B/A at 10% peak height are recommended.
The current problems with the LOD concept in chromatography are
reviewed. They include confusing the LOD with other concepts in trace
analysis; the use of arbitrary, unjustified models; the use of
concentration units instead of units of amount; and the failure to
account for differences in chromatographic conditions (bandwidths) when
comparing LODs.
Two models are proposed for calculating the chromatographic LOD. A
new concept, the standardized chromatographic LOD, is introduced to
account for differences in chromatographic bandwidths of experimentally
measured LODs. The standardized chromatographic LOD is shown to be a
more reliable CFOM than the conventional (nonstandardized)
chromatographic LOD.
Vll
CHAPTER 1
INTRODUCTION
Overview
Chromatography is a wellknown method for the separation and
quantitation of chemical moieties from a (sample) mixture. Over the
years several concepts and parameters, hereafter referred to as
chromatographic figures of merit (CFOMs), have been introduced to
characterize the separation and quantitation. Unfortunately, some of
the CFOMs are often difficult to estimate [those which characterize
chromatographic peaks]; others are ambiguous [e.g., the limit of
detection (LOD)]. The goal of the present work, which is introduced in
more detail in the following two sections, is the improvement of the
measurement and interpretation of these chromatographic figures of
merit.
It is beyond the scope of this work to introduce or review the
development of these CFOMs from either a historical or theoretical point
of view. Such discussions and references to additional discussions may
be found elsewhere (110).
Chromatographic Peak Characterization
In recent years there has been considerable interest in the
characterization of experimental chromatographic peaks. Presented in
Table 1.1 are the names, symbols, and general expressions that have
evolved for the parameters used in chromatographic peak
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characterization. The graphical chromatographic parameters are
illustrated in Figure 1.1.
These CFOMs have been estimated either manually using graphical
measurements made directly from the chromatogram or by a computer
following data acquisition. Both methods have advantages and
disadvantages.
Manual methods were used exclusively at first and are employed
quite extensively today. For arbitrary peak shapes, they are accurate
for only five CFOMs: t fi , B/A, h p , W b , and W . If a Gaussian peak shape
is assumed, however, then M^ = t R , and Mp is only a function of W,, W ,
or Mq and N may subsequently be calculated. Except for higher
even central moments, the remaining CFOMs are zero for Gaussian peaks.
For real chromatographic peaks, it is almost always a mistake to
assume a Gaussian peak shape. Experimentally these ideal, symmetric
peaks are rarely, if ever, observed due to various intracolumn and
extracolumn sources of asymmetry (5,1123). Kirkland et al. have shown
that the plate count can be overestimated by as much or more than 100%
if any of the three most common Gaussianbased equations are
employed (23).
Computer estimation methods are more accurate than common manual
methods for a given CFOM but are not available to every chromatographer.
The general approach taken has been one of peak statistical moment
analysis (6,11,2224). Via relatively simple algorithms all the CFOMs
may be determined quite accurately, though the precision of the second
and higher central moments is seriously affected by baseline noise (25).
The failure of the Gaussian function as a peak shape model for real
chromatographic peaks led to the search for a more accurate model and
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fraction a = 0.10. Except for A, B, and B/A at a = 0.10,
all width related measurements are subscripted with the
value of a to prevent ambiguity.
the eventual acceptance of the exponentially modified Gaussian (£MG), a
function obtained via the convolution of a Gaussian function and an
exponential decay function which provides an asymmetric peak profile.
The development, characterization, and theoretical and experimental
justification of this model have been thoroughly reviewed (21,22,26,27).
Previous chromatographic studies (11,12,14,15,1723,2531) involving the
EMG function, summarized in Table 1.2, demonstrate the utility of this
skewed peak model.
Adoption of the EMG peakshape model has improved the estimation of
the CFOMs. A new algorithm for the computerbased peak moment analysis
has been derived (25) and tested (22) which is less sensitive to
baseline noise and the uncertainty of peak start/stop assignments. More
recently, Barber and Carr described a manual method for CFOM
quantitation which requires the graphically measurable retention time
tpi peak width W, empirical asymmetry factor B/A, and successive
interpolations from three largescale universal calibration curves
(31,32).
The primary objective of this part of the present study is the
development, using the EMG model, of accurate equations for CFOM
calculation dependent solely on tp, Wq « , and B/A. The need for
computerized data acquisition is thus circumvented, and, in addition,
CFOM calculation via these equations is expected to be faster and more
precise than the other accurate manual method since no graphical
interpolation is required.
The previously reported methods for evaluating the EMG function
(14,16,19,27) and obtaining chromatographic peak data (19,26,31) were
too inaccurate or too unwieldy to use in the present study, which
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employed an Apple II Plus microcomputer. This necessitated, therefore,
the development of a simpler method for evaluating the EMG function and
the incorporation of a simpler, more accurate, and more general set of
algorithms for obtaining the EMG data of interest.
Limit of Detection
The limit of detection (LOD) is generally defined as the smallest
concentration or amount of analyte that can be detected with reasonable
certainty for a given analytical procedure. Though arguably the most
important figure of merit in trace analysis, the LOD remains an
ambiguous quantity in the field of chromatography. Detection limits
differing by orders of magnitude are frequently reported for very
similar (sometimes identical!) chromatographic systems. Such huge
discrepancies raise serious questions about the validity of the LOD
concept in chromatography.
The primary objective of this part of the present work is to
restore the integrity of the LOD concept, to make the chromatographic
LOD a reliable, meaningful figure of merit. This will be accomplished
in two steps: First, the major sources of the discrepancies in
chromatographic detection limits, i.e., the current problems with the
LOD concept, will be identified. Second, each problem will be addressed
and eliminated (or circumvented).
CHAPTER 2
GENERATION OF THE EXPONENTIALLY MODIFIED GAUSSIAN (EMG) FUNCTION
AND RELATED DATA
Introduction
This chapter describes the improvements achieved by this study in
evaluating the EMG function and in obtaining the EMG data of interest.
The results of the present work are compared with those previously
obtained. In addition, the corresponding BASIC computer program, EMGU,
is listed and discussed briefly in Appendix A. Universal EMG data are
tabulated in Appendix B. It is hoped that this will facilitate the use
of the EMG function, whenever applicable, in modeling studies in
chromatography or any other area.
EMG Evaluation
Background
Description of the EMG function . It is beyond our scope to derive
the EMG function from first principles. Those who so desire should see
the treatments given by Sternberg (11, pp. 250253) or Kissinger et al.
(12, pp. 159162). Their results are shown below in eq 2.1.
z/(2) 1/2
*EMG XW = *i\UQ{£) /ij expLu.muQ/ T ; _ ^ o — c G , v '
h £MG (t) = [Atf G (2) 1/2 A] exp[0.5(a r /T) 2  (tt r _)A] /exp(x 2 )dx (2.1)
/•
PE 1 E 1 I 1
where z = (t  t G )/a Q  On/x . (2.2)
Equation 2.1 shows that the EMG function is defined by three parameters:
10
the retention time, t Q , and standard deviation, a Q , of the parent
Gaussian function; and the time constant, t ? f the exponential decay
function. In addition, the quotient t/Oq is a fundamental measure of an
EMG peak's asymmetry. The arbitrary constant A determines the amplitude
of the function. Note that the right hand side of eq 2.1 can be broken
into 3 parts: the preexponential term (PE.,), the exponential term
(£.]), and the integral term (I,).
Discrepancies. A literature survey which I conducted revealed
three discrepancies in eq 2.1. In the first instance, a factor of 2
difference observed in the preexponential term, PE,, is relatively
unimportant because this affects only the zeroth statistical moment
(peak area). All other parameters, including higher order moments which
are normalized by the zeroth moment, are unchanged by this factor of 2.
The second and third discrepancies are serious, however. Both were
observed in the denominator of the second quotient, (t  t Q )/T, within
the exponential term, E.,. In one case, an additional factor of 2 was
present; in the other, a Q was added to this denominator and omitted from
the numerator of the preexponential term, p£,. These errors invalidate
those expressions for the EMG function.
Evaluation of the Integral Term
Range of z. The methods used to evaluate eq 2.1 have frequently
been omitted from the EMG literature. Since the evaluation of the first
two terms is straightforward, the reported methods differ only in the
manner in which the integral, Ij, is determined; the Ij approximations,
in turn, depend on the value of z in the upper limit of I.. It is
convenient to group the range of possible z values into three regions:
a) z <. 3; b)3 <. z < 4; and c) z >. 4 . In region c, the definite
11
integral equals the constant (tt) 1/2 to within 0.01? and is thus
virtually independent of z. In regions a and b, however, the relative
value of the definite integral is highly dependent on z. Numerous
techniques permit the accurate evaluation of L in region b. In region
a, however, as z becomes more negative, it becomes more difficult to
approximate the integral to the same (high) relative accuracy. Thus it
is pertinent to examine the practical minimum values of z for EMG peaks.
As seen from eq 2.2, z depends on a) the normalized difference
between the time of interest, t, and t Q ; and b) the reciprocal of the
peak asymmetry, (T/a G )" 1 . For a given peak shape (constant t/o g ), z
will therefore be smallest (most negative) at the starting threshold of
the peak, which may be conveniently defined as that time (t) on the
leading edge of the EMG peak where the value of the function Ch Pun (t)]
is a specified fraction, B, of its maximum value, i.e., h EMG (t)/h p = B.
Moreover, the minimum z value will decrease as the starting threshold is
decreased. This is shown in Table 2.1 where minimum z values are
tabulated for EMG peaks with asymmetries ranging from 0.1 to 3. Another
trend illustrated is that for a given starting threshold, z ■ increases
with increasing asymmetry (Va G ).
Previous methods. Given this wide range of z values, how is I 1
(in eq 2.1) evaluated? Except for a vague reference to an unspecified
polynomial approximation (30), all previous methods for calculating I 1
employ different techniques for different values of z. For moderate z
values (e.g., regions b and c, above), some methods utilized the well
known identity
/.
exp(y 2 )dy = 0.5U) 1/2 [1 + erf(x)] (2.3)
12
Table 2.1. Minimum z Values Needed to Evaluate EMG Peaks for
Various Asymmetries (r/a Q ) and Starting Thresholds (B) a
0.001 0.01 0.1
T/0 G
0.10 b 12.9 12.1
0.15 10.3  9.6  8.7
0.20  8.6  7.9  7.0
0.25  7.5  6.9  6.0
0.30  6.8  6.2  5.3
1.00  4.4  3.6  2.7
3.00  3.6  2.8  1.9
a See eq 2.2 and text for description of z, T/a Q , and B.
Values for z have been rounded.
b An underflow error occurs z < 13, thus preventing its
measurement.
13
(where erf is the error function) in order to take advantage of error
function subroutines resident in the computers used (19,27). Others
used eq 2.3, but approximated the error function by interpolation from a
set of tabulated areas (14,18). For small z values (region a, above),
the error function techniques were sufficiently inaccurate to warrant
the use of other methods instead. In nearly every instance some type of
asymptotic series was employed (14,19,27). In the lone exception, a
Gaussian function was substituted for the EMG function for any part of
the peak profile where very small z values were encountered (18).
This work. The approach taken in this study is to transform the
integral in eq 2.1 via change of variable [x = y/(2) 1/2 ] to
(tt) 1/2 /exp(y 2 )dy/(27r) 1/2
The EMG function can now be written as
h EMG (t) = tAa G (27T) 1/2 / T ][E 1 (see eq 2.1)] / exp(y 2 )dy/(2Tr) 1/2 (2.4)
■ /co
?E 4 * h "
The integral in eq 2.4 can be approximated by a polynomial approximation
l 4 (z<0) = NF(z) P(q) and l 4 (z>.0) = 1 l 4 (z<0), where
NF(z) = exp(z 2 /2)/(27r) 1/2 , P(q) = ^q + b 2 q 2 + b 3 q 3 + b^q 4 + b 5 q 5 ,
•1
q = (1 + pz)~ , and p,b 1 ,...,b 5 are constants given in Table 2.2 (33).
Comparison. The values obtained for 1^ are compared to the true
values (34,35) in Table 2.3 from z = 10 to z > 3.9. In addition, since
In = ^/(tt) 172 (2.5)
they can be compared to values obtained for I* via an asymptotic
14
Table 2.2. Constants in the Polynomial Approximation
for lu in eq 2.4 a
p = 0.2316419 t> 3 = 1.781477937
b 1 = 0.319381530 b 4 = 1.821255978
b 2 = 0.356563782 b 5 = 1.330274429
a See text immediately after eq 2.4 or reference 33.
15
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16
series (27), the most accurate method reported for z < 3. (Comparison
with values for I 1 obtained via error function techniques when z > 3
(Ijj > 10"°) is not illustrative since the maximum absolute error in the
polynomial approximation Iu(z) is estimated to be + 7.5 x 10~ 8 .)
Table 2.3 shows that our method for evaluating 1^ is exceptionally
accurate for moderate values of z. Moreover, it compares favorably with
the asymptotic series method for evaluating I,, except for z < 8 where
the latter method is somewhat better. Reexamination of Table 2.1
shows, however, that for all practical purposes z > 8 whenever T/a G J>
0.2. Thus our method for evaluation of 1^ can be used to evaluate the
EMG function to within 1% or less for i/o n > 0.2.
While being slightly less accurate than the most accurate previous
method, the new technique for evaluating the EMG function is much more
convenient than any of the previous methods. Only one simple subroutine
requiring just a few programming lines (see lines 29903160 in Appendix
A) is needed, whereas the other methods require at least two
subroutines, if implemented on any computer without a builtin error
function routine (i.e., nearly all microcomputers and many
minicomputers) .
Obtaining Universal EMG Data
Background
Using the polynomial approximation for the integral in eq 2.4, the
EMG function can be evaluated over the entire practical time range.
Depending on the data required for the modeling process of interest, the
EMG peaks could be generated "on the fly" as needed. Alternatively, the
necessary EMG data could be generated (and stored) in advance and
accessed when needed. Though the storage requirements may seem
17
prohibitive for the latter, given a T/a Q value and a peak height
fraction, a, three quantities completely specify an EMG peak (31).
Figure t.1 shows an EMG peak with its pertinent graphical parameters.
Regardless of the retention time, t Q , and standard deviation, o of the
(unconvoluted) parent Gaussian peak (not snown) , (B/A) . W_/a P . and (t D 
^G^ a G are universal constants so long as T /° G and a remain fixed.
Recent work has utilized these universal data sets almost exclusively
(3D.
Experimentally, three parameters must be determined in order to
calculate the universal data: t R , t«, and t B (see Figure 1.1). Note
that t R must be obtained before t. or t g because h p = h EMG (t R ) is needed
for the latter.
Measurement of the Pertinent Peak Parameters
Previous methods . In the past, t R nas been determined by one or
more of the following methods: a) peak displacement data and knowledge
of t G (19); b) differentiation of h £MG (t) and solving for roots (26);
and c) least squares fitting of the top of the peak with a quadratic
gram polynomial (26,31).
Once t R has been found, t A and tg can be located. In the only
method reported previously, two points [t.,, hg^tt^], [t 2 , h EMG (t 2 )]
are found so that t 1 < t A < t g (or t 1 < to < to). Linear interpolation
yields the approximation for t« (or t B ).
This work . The approach for finding these quantities is based on
iterative search mechanisms, as the flowchart in Figure 2.1 shows. In
the case of t R , initial time limits are easily found using the fact that
t R is always greater than t Q . The EMG function is then evaluated from
18
Find initial time limits
Y
J±.
Calculate new time limits
N
t = t + dt (t R , t B only)
t=tdt (t A only)
Y
>
Calculate value of t (and h )
f A° r V
N
_^L
Decrease value of dt
Figure 2.1. Simplified flowchart for locating t (and h ), t , or t .
The El'IG peak parameters must be input before beginning this
search.
19
the lower time limit to the upper time limit in increments of dt. When
the maximum is found, new lower and upper time limits [given by
t( current)  2dt and t(current), respectively] closer to the peak
maximum are set, the time increment is decreased, and the search is
begun again. The retention time, t R , is approximated by t(last) 
dt . .
mm
The algorithm for estimating t A (or tg) is similar to the t. (or
t B ) search algorithm previously discussed in that the time limits are
analogous. Since t A < t R < tg, initial values for t 1 and to are easily
determined. To locate t A (or tg) , the EMG function is then evaluated
from t s tg to t^ in decrements of dt (or from t 1 to to in increments of
dt) until h EMQ (t) < ahp. New values for t^ and to [given by t( current)
and t(current) + dt, respectively] closer to t A (or tp) are then set,
the time decrement (increment) is decreased, and the search is repeated.
This is continued until t< and t~ are known to the desired precision; t»
(or tg) is then given by (t 1 + tg)/2.
The maximum error in the values of t R , t A , tg and related universal
EMG quantities obtained via these search algorithms is presented in
Table 2.4. In all cases the error is dependent on the smallest (most
precise) time increment (or decrement), dt ^ i used in the last
iteration of each search. In theory, dt min could be as small as
desired. Due to the finite precision of computers, however, the time
increment, dt , would ultimately be reduced to such a low value that
h EMG (t ) ~ h EMG (t+dt). (2.6)
Henceforth the algorithms would cease to function accurately, if at all.
20
Table 2.4. Maximum Errors in the Universal
EMG Data and Selected Component Parameters
Parameter 3 Maximum Error
H ± dt min
*!• fc B ± 1/2 dt min
A = t R t A , B = t B t R ± 3/2 dt m . n
V°G = (V t A> /a G ± dt min /a G
( W /ff G ± dt min /G G
a A, B, to » t., and tg defined in Figure 1.1;
a,,, t P , and dt .„ described in text.
G' U' mm
21
As dt is decreased, the t R search algorithm will fail first, since
the slope of the EMG function is smallest in the region of t Q . Somewhat
smaller (more precise) dt min 's could be employed in the searches for t.
and tg before the algorithm breakdown described by eq 2„6 would occur.
The increase in precision of t A and t fi is probably not worth the effort,
however, since two out of the three universal EMG data expressions are
dependent on the least precise quantity, t R .
The minimum usable value of dt depends on the precision of the
computer employed and on the value of o Q chosen. In this study, the
experimentally measured minimum ratio of dt/ov, was 0.0002 for 0.1 < t/o q
< 3 using single precision arithmetic. Multiple precision capabilities
would allow a still lower dt/cr„ ratio to be used.
Comparison . The algorithms for t R , tj,, and tg may be compared as
follows :
1. With the exception of the quadratic least squares method for finding
tg, all of the methods for obtaining t R , t,, and tg are designed for
simulated data (essentially no noise).
2. The two algorithms for calculating t. (or tg) are quite similar.
Both require two points which closely bracket the desired peak height
fraction, a, and both are relatively unbiased. The subsequent
interpolation performed in the previously described algorithm is
potentially more precise than the averaging of the final time limits in
the proposed search algorithm. If t» and tg are already known as
precisely as or more precisely than t R , however, additional improvements
in their precision, even if realized, will not yield significant
increases in the precision of B/A and (t R t G )/ a G .
3. Our approach for determining t R , though crude, is superior to the
other three methods previously discussed for the following reasons:
i) It is a general algorithm. Whereas methods a and b are specific to
the EMG model, our search mechanisms will work for that and other peak
models as well.
ii) It is accurate and unbiased. In contrast, the quadratic least
squares fitting method, though general, suffers from a small, but
nevertheless observable bias (due to a determinate error) which
increases with increasing peak asymmetry (26).
iii) It is easy to understand and implement. The other methods are
unnecessarily complex, though they admittedly have the potential
for greater precision.
22
4. The proposed search algorithms for determining t R) t A , and to are
superior to the previous methods because they can be debugged more
easily. By having the time and the value of the EHG function printed
every time the EMG function is evaluated, the programmer can literally
watch the computer perform the search. Since the search logic is so
simple, programming errors are easily detected. Upon elimination of the
errors, the print statement may be removed.
Comparison of Universal EMG Data
Table 2.5 shows representative sets of universal EMG data obtained
from this study and from a previous work (32) which utilized the
quadratic least squares method and the interpolation method for the
location of t R and t A (or t g ), respectively. The precision is reported
for our data (in terms of maximum errors) and is assumed to be no worse
than + 1 in the least significant digit of the previously reported data.
Several points should be noted:
1 . Although this difference is slight or nonexistent at high
asymmetries, the previously obtained universal data are somewhat more
precise. This is expected since the algorithms used in locating t R , t.,
and t B are potentially more precise than those developed here.
2. The data sets are in excellent agreement for all three universal EMG
quantities at low asymmetries (t/ct„ <. 0.5). This agreement is
especially significant at i/o Q =0.1, because it shows that the moderate
errors introduced by the polynomial approximation for K in eq 2.4 when
12 < z < » (see Table 2.3) are not transmitted to the universal EMG
data.
3. Whereas the W a /a Q data reported previously are consistent with the
corresponding data of this study over the entire range of T /cr p , the
remaining data sets are discordant for x/o Q 2 1.0. Relative to the
current data sets, the previous ones for (t R t G )/a Q and (B/A) appear to
be slightly overestimated and underestimated, respectively. This
discrepancy is due to the use of a least squares fitting method in the
previous measurement of t R which overestimates this quantity for EMG
peaks (26) and other types of skewed peaks. This bias increases from an
insignificant value at low x/a Q to an observable one at r/a _> 1.0,
4. Despite the differences noted above, the general interlaboratory
agreement is quite good. Though the current data are more accurate,
either set of EMG data can be used with confidence for modeling studies.
23
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Conclusion
Though the exponentially modified Gaussian (EMG) model has already
been employed in numerous studies in chromatography as Table 1.2 shows,
its usage might have been still more extensive had it not been for some
confusing discrepancies and for the overly complex methods used for its
evaluation reported previously. Hopefully the clarification of these
discrepancies and presentation of a simple method for evaluating the EMG
function and obtaining universal data will encourage more scientists in
all fields to use the EMG model, when appropriate.
a
CHAPTER 3
EQUATIONS FOR THE CALCULATION OF CHROMATOGRAPHIC
FIGURES OF MERIT FOR IDEAL AND SKEWED PEAKS
Introduction
Adoption of the EMG model for chromatographic peak characterization
results in a new set of chromatographic figures of merit (CFOMs) which
re listed in Table 3.1. These CFOMs consist of fundamental and
derived EMG parameters, the latter containing explicit expressions for
the first through fourth statistical moments defined previously in
Table 1.1. Included among these new CFOMs are the following: the
retention time, t Q , and standard deviation, a Q , of the associated parent
Gaussian peak from which the skewed peak is derived; the exponential
modifier, t; the fundamental ratio, t/Oq, which characterizes peak
asymmetry; the observed efficiency (number of theoretical plates) of a
given (asymmetric) chromatographic system, N ; the maximum efficiency
a given system could achieve, N max , if all sources of asymmetry were
eliminated; and finally, two CFOMs proposed originally in this work
which demonstrate how peak asymmetry drastically reduces chromatographic
efficiency— the relative system efficiency, RSE, and the relative plate
loss, RPL.
Derivations
If the estimate G obtained from using a Gaussian peak shape
equation is used to approximate the true value T of a CFOM for an
asymmetric peak, the relative error RE which results is defined as
25
26
Table 3.1. Chromatographic Figures of Merit Based on
the Exponentially Modified Gaussian (EMG) Model
Fundamental
t , o Qf t, x/a G
Derived
N sys = H 2 '(% 2 + t2 )
N max ~~ (V a G )£
RSE = ^ T sys /N max) <W 2 = ° G 2/M 2
RPL = C(N max  N sys )/N max ] 'W* = 1 " RSE = t2/M 2
M = t Q + x
M, = r 2 + t 2
M 3 = 2 T 3
M 4 = 3ct q 4 + 6a Q 2 t 2 + 9 t 4
Y s = M 3 /M 2 3/2
f E = M 4 /M 2 2  3
27
RE = (G  T)/T (3.1)
which can be rearranged to give
T = G/(fiE + 1) (3.2)
Thus, the true value T and the Gaussian approximation G for the CFOM are
related by the correction term (RE + 1) in the denominator of eq 3.2.
Kirkland et al. have shown that
RE = f(T/a Q ) (3.3)
for the three popular Gaussianbased methods for determining plate
counts of a system (23). Equation 33 should hold, in fact, for any
CFOM for which a Gaussian approximation exists except t Q . Since Barber
and Carr have shown (31) that
t/a G = f(8/A) (3.4)
successive substitution of eqs 33 and 3.4 into eq 3.2 yields
T = G/[f(B/A) + 1] (3.5)
Although the exact form of f(B/A) is unknown, a least squares curve
fitting of an RE versus B/A plot can give an excellent approximation.
The above approach was used for calculation of N , a rt and M .
sy s u c.
Following this, t was calculated (see sixth equation, Table 3.1) by
x . [M 2  a Q 2]V2 (3<6)
For the determination of t Q , the universal relationship
(t R  t G )/ff Q = f(T/a Q ) (3.7)
previously reported (31) was rearranged and combined with eq 3.4 to give
fc G = H ~ a G f(B/A) (3.8)
28
where f(B/A) was approximated by a least squares fit of (t R  t Q )/a G
vs. B/A.
Since t R , B/A, and W 0>1 are graphically measurable, all the
remaining CFOMs can be calculated once the fundamental parameters o„, t,
and tn have been determined.
Experimental
Apparatus
An Apple II Plus 48K RAM microcomputer was programmed in BASIC for
EMG peak generation. A curvefitting program available from Interactive
Microware (P.O. Box 771, State College, Pa., 16801) with linear,
geometric, exponential, and polynomial capabilities was used for the
unweighted least squares fitting of various data sets.
Procedure
EMG Peak Generation. Except as otherwise noted, values of A = 1,
t G = 100, and ° G = 5 were used in eq 2.4 for EMG peak generation. The
x /°q ratio was varied from 0.1 to 3 in 0.05 increments producing data
equivalent to 59 peaks. The times for t R , t A , and tg (see Figure 1.1)
at a = 0.1, 0.3, and 0.5 for each of the 59 peaks were determined to
within 0.001, using the simple search algorithm described in Chapter 2;
corresponding values of W and B/A were then computed.
Development of the CFQM equations . Textfiles of fiE(N , M , a r )
vs. (B/A) 0<1)0>3j0>5 , (t R  t G )/° G vs. (B/A) 0#1>0#3)05 , and RSE vs.
^ B/A ^0.1 ,0.3,0.5 were made for ° 1 i T/cr G ^ °'3 The Gaussian relations
CT G = w 0. i/' 4  2 9 1932 and M 2 = (W Q# ^4 .291932) 2 were used in the Gaussian
approximations for ^ sy3 , a G , and M 2 at a = 0.1. Similar equations were
used for M 2 an d a Q at a  .3, 0.5. The true values of N gys , a Q , M 2 ,
29
and RSE were computed from the known values of x, o«, t Q , and t R for a
given peak.
The CFOM equations were developed by unweighted least squares curve
fitting. The relationship between several quantities [e.g., RE(N),
(B/A) a , to a / CT Q] and t/o q has been shown previously (23,31) to be
nonlinear for < V° G < 0.5 and nearly linear for 0.5 <. T /°n <. 3.
Because similar relationships were observed in some of the textfiles
above, least squares fitting was limited to about the same t/cu range
0.53 (1.09 < B/A <_ 2.76) except for the RSE vs. B/A textfile where
the complete set of values was used in the regression analysis.
Although the least squares fittings of the various B/A textfiles were
initially judged by visual inspection and the coefficient of
determination (square root of the correlation coefficient), their final
evaluation was based on the accuracy and simplicity of the resulting
CFOM equations.
For ease of use, the CFOM equations were first simplified
algebraically and then by the successive rounding of numerical
coefficients. The occasional nearness of the decimal coefficients to
whole numbers was exploited. For example, if f(B/A) = 1.02(B/A) + 0.69,
then for 1 .09 <. B/A i 2.76, the much simpler function
f(B/A) = B/A + 0.72 is approximately the same (exactly if B/A = 1.5) and
the accuracy of this function is not significantly affected.
All CFOM equations were simplified as much as possible — any more
rounding of the coefficients will result in appreciably greater error.
Evaluation of the CFOM equations . The accuracy of the CFOM
equations was evaluated in terms of four parameters, listed in
decreasing order of importance: the percent relative error limits
30
($RELs) which represent the maximum possible error of the CFO'M equations
within the specified B/A range; the mean percent relative error or bias,
J&RE = E%RE/n; the average magnitude of the percent relative error,
Z$RE/n; and the standard deviation of the percent relative error.
The precision of the empirical, EMGbased CFOM equations and three
Gaussian CFOM equations was calculated via error propagation theory and
is reported as percent relative standard deviation (S&RSD). The
required precision estimates of t R , W a , and (B/A) (the graphically
measurable quantities) were obtained using data from a previous
study (32).
Results
Recommended CFOM Equations
Listed in Table 3.2 are the empirical CFOM equations based on t R ,
W 0.1' and B/ ^ A measurements (see Figure 1.1) which we recommend.
Accuracy. Using equation 1 in Table 3.2, the true efficiency of a
chromatographic system, N sys , may be estimated to within + 1.5$ for both
Gaussian and exponentially tailed peaks within the asymmetry range 1.00
1 B/A i. 2.76. Equations for Mg, t Q , and M 1 are equally if not more
accurate over this asymmetry range.
All CFOMs except RPL, Mg, Y g> and Y E can be estimated to within
± 5% for 1.09 <. B/A <. 2.76, the asymmetry range over which most of the
curve fitting was performed.
All CFOM equations are accurate to within + 5% for 1.19 <. B/A <_
2.76, and 18 out of 21 are accurate to within ±2%.
31
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Except for eqs 11b, 13b, and 14b which had biases of +0.976$,
+0.820$, and +1.111$, respectively, the bias for every equation given in
Tables 32, 33, and 3.4 was less than + 0.6$.
Precision Summary. For the equations in Table 3.2, the estimated
relative standard (RSD) limits obtained via propagation of error theory
for N sys' M 2' a G' N max' and RSE were a11 less than or equal to + 4.5$.
RSD limits for t« and H< were + 0.2$.
The precision of the EMG equations in Table 3.2, the widthbased
Gaussian equations, and the calibration curve method of Barber and Carr
(3D is compared for N sya , M 2 , and a Q in Table 3.5. The results shown
for the Gaussian equations are valid for 50$, 30$, and 10$ width
measurements because RSD(W 0>5 ) = RSD(W Q>3 ) = RSD(W Q 1 ). The slightly
greater imprecision observed for the EMG equations is due to uncertainty
in the B/A measurement not required for the Gaussian equations. The
somewhat larger $RSDs for the method of Barber and Carr are probably due
to interpolation uncertainties (from the calibration curves) unique to
this method.
The precision of the remaining CFOMs in Table 3.2 was found to be
highly dependent on the peak shape. Rather than reporting RSD limits,
the RSDs for several CFOMs or groups of CFOMs have been plotted vs B/A
in Figure 3 1 •
Other CFOM Equations
Listed in Tables 3.3 and 3.4 are smaller sets of CFOM equations
developed only for use in determining if a real chromatographic peak is
wellmodeled by an EMG peak and should not be used to routinely
calculate any CFOM (including N ) since they are usually less
accurate, less precise, and more complex than the analogous equations in
34
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Table 3.5. Precision 3  Comparison of Three Graphical Methods for
Estimating N , M 2 , and ° G
method
Gaussian eqs
sys
+ 2.0
M
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empirical eqs,
Table III,
this work
+ 2.5
+ 2.4
± 2.0
calibration curve
method, reported
previously (3D
+ 5
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± 3
a,.
Reported as percent relative standard deviation (%RSD).
Precision of equations estimated via error propagation, usins
data of Table 3.6.
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39
Table 3.2. However, the accuracy and precision for MU, a Q , t Q and t at
a = 0.3 and 0.5 are still sufficient to permit peak modeling decisions
to be made.
Discussion
Detailed Discussion of Precision
Shown in Table 3.6 are the precision data for t p , W , and (B/A)
It EL 3,
used in this study. The RSD results were obtained by converting
previously reported raw precision data (32) to the form appropriate for
error propagation analysis for the conditions specified in Table 3.6.
Data excluded in the previous study for (B/A  1) were also excluded in
CI
this analysis. The RSDs of t R , W & , and (B/A) for individual peak
shapes (at a given peak height fraction) were averaged as done
previously, thereby implicitly assuming the independence of the RSDs on
peak shape.
The RSDs for t R , W Qf and (B/A  1 ) for individual peak shapes were
originally reported relative to a Q , a Q , and (B/A  1) , respectively.
Multiplication by o Q /t R , ° Q /^ a , and (B/A  1) /(B/A) converted them to
the appropriate form.
Since the RSDs of t R , W & , and (B/A) were assumed to be independent
of peak shape, intuitively it might seem that this should also be true
for the RSD of any calculated CFOM. This is not the case, however. For
one group of CFOMs (N sys , M £l o Q t Q , N max> RSE, and M,), a slight to
moderate variation in their RSDs with (B/A),, was observed. This can be
explained by examining the random error propagation in the general
empirical N equation,
N sys = C 1 (t R /W a ) 2 /[(B/A) a + C 2 ] (3.9)
■
40
Table 3.6. Suggested Chromatographic Measurement Conditions and
the Resulting Precision (%RSD) Achieved for t„, W , and (B/A)
n a a
Conditions
1. Chroma togram recording rate: 1 cm/o G (W Q < _> 4.3 cm)
2. Ruler resolution: +0.2 ■
3. Minimum retention distance, (t R ) . : 10 cm
4. Minimum peak height, (hp) min : 10 cm
Results a ' b
CFOM (a = 0.1) (a = 0.3) (a = 0.5)
fe R +0.2 identical identical
w a ± 1 ° ± 1.0 + 1.0
(B/A) +2.0 +2.5 +3.0
Data obtained from reference 32 and subsequently converted (see
Detailed Discussion of Precision) for a = 0.1, 0.5 — results
interpolated for a = 0.3.
%RSD(t R ) rounded to nearest 0.1%; %RSDs for W , (B/A) rounded
to nearest 0.25%.
41
Assuming negligible covariances, RSD(N ) is given by
sys
RSD(N sys ) = {4[RSD(t R )] 2 + 4[RSD(W a )] 2 +
1 2
RSD[(B/A) a ] 2 [(B/A) a /((B/A) a + C 2 )] 2 } 1/2 (3.10)
3 4
Even when terms 13 in eq 3.10 are constant, RSD(N „) will vary
sys
somewhat with (B/A) a because of term 4. Clearly this variation will be
greatest for (B/A) a < C 2 < 0. Additionally, as C 2 ■+ (B/A) ,
RSD(N gys ) *• ». For C 2 =0, RSD(N sys ) is essentially independent of
(B/A) a . Finally, for C 2 > 0, a negligible to slight variation of
RSD(N s ) with (B/A) a may be observed, depending on the magnitude of
terms 1 and 2 relative to the product of terms 3 and 4. Except for eqs
1 and 2(a,b) in Table 34, the RSD limits for N , M OJ a r , t P , N
sys ' d ' h ' b ' max '
RSE, and M., calculated via equations in Tables 3.2, 3.3, and 3.4 varied
by less than 0.5% for 1.00 <. B/A <. 2.76.
The remaining CFOMs (t, t/o q , RPL, M,, M 4 ,Y s ,Y e ) comprise a second
group whose RSDs are moderate to strong functions of peak shape as
Figure 31 shows. In every instance the imprecision is largest for the
least asymmetric peaks and smallest for the most highly skewed peaks.
Analysis of the error propagation equations show that one or more terms
within the equations get very large as the peak shapes become symmetric.
Why Measure at 10% Peak Height?
For reasons listed below, the recommended CFOM equations in
Table 3.2 are based on (in addition to t R ) the measurement of W and B/A
at 10% peak height rather than at other peak height fractions such as
50%, 30%, or 5%:
•*—■
42
1. Examination of Tables 3.2, 33, and 3.4 shows that many CFOM
equations at 10$ are olearly superior to the corresponding ones at 30$
and 50$ in terms of
a) precision (lowest RSD limits)
b) widest working range for equivalent accuracy (e.g., N )
c) simplicity for M~ (path a)
2. The N s equation at 10$ peak height is more accurate for Gaussian
and nearGaussian peaks than other N equations developed at 50$, 30$,
or 5$. For example, at B/A = 1 the relative error was +0.6$, +10.0$,
+2.0$, and +2.5$, respectively.
3. In a previously reported graphical measurement study (31,32),
statistically significant positive and negative biases were detected in
the measurement of A Q 5 and B Q 5 , respectively, resulting in a
consistent underestimation of (B/a) q 5< No such biases were detected
for Aq ^ and B Q «, and only a slight underestimation was observed for
(B/A) Q> J.
4. It is likely that RSD(W 0#05 ) > RSD(W 0<1 ), since in going from W Q 1 to
W 0.05 the ma S ni tude of the slope of the peak (on either side) decreases
much more rapidly than the peak width increases. Thus the precision for
the 5$ CFOM equations would be poorer [assuming RSD(Wq Q5 ) contributes
substantially to the total uncertainty].
5. Superior resolution between overlapping peaks is required for
measurements at 5$ peak height than at 10$.
6. As exemplified in Figure 3.2 for N , Gaussian CFOM equations based
on width measurements at 10$ are much less inaccurate (though still
exceedingly in error) for asymmetric peaks than the corresponding
Gaussian equations at 50$ (shown) and 30$ (not shown). That is, the
slope of the RE vs. (B/A) a (shown for a = 0.1) plots is smaller; thus
the approximate RE correction function in the denominator of eqs 3.2 and
3.5 (text) will be less sensitive to the measurement imprecision of
(B/A) a ).
7. The sensitivity of the relative error (RE) correction functions to
the (B/A) a measurement imprecision is only slightly lower at 5$ than at
10$ peak height (see Figure 3.2) and is insufficient to warrant CFOM
estimation at 5$.
6. As seen in Table 3.7, the RSE can be calculated much more accurately
using width measurements at 10$ than at 50$, 30$, or 5$. In addition,
the precision is much better (lower RSD limits) at 10$ than at 50$ or
30$, and is comparable to that at 5$.
9. The empirical asymmetry factor measurement, B/A, was introduced at
10$ peak height rather than at 50$ or 30$ because peak tailing is much
more apparent at 10$. Since then almost all empirical measurements of
asymmetry have been reported at this peak height fraction; these data
will be of little value in later years if the B/A peak height fraction
is redefined.
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Table 3.7. Comparison of the Accuracy and Precision of the
fiSE Equations at a = 0.05, 0.10, 0.30, and 0.50
equation
RSE = 1.04 [(B/A) 0#05 r 2  00
RSE = 0.99 C(B/A) Q#10 ]
2.24
RSE = 0.926[(B/A) 0>30 ] 3 ' 11
RSE = 0.913[(B/A) 0>5Q ] 4  33
#RE limits
% RSD
5.0, +4.5
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"
46
10. It is easier to mentally compute \Q% of an arbitrary peak height
than 50%, 30%, or 5%.
Taken collectively, the above arguments indicate that the best CFOM
estimation is obtained from graphical chromatographic measurements at
10% peak height.
General Aspects
Preliminary mode ling of experimental peaks . Chromatographic peaks
should be examined for their resemblance to Gaussian, EMG, or other peak
shapes, first by visual inspection and then from the asymmetry factor
measurement. In the unlikely event that B/A = 1, the validity of the
Gaussian model can be checked by comparing the measured peak width
ratios W Q>5 :W q.3 :W 0.1 to the theoretically predicted ratios 0.5487 :
0.7231 : 1. For B/A 2 1.09, the validity of the EMG model can be judged
by the agreement of values of o Q , M 2 and/or x, and t Q determined from
both B/A and W a measurements at a = 0.1, 0.3, and 0.5 (see Tables 3.2,
3.3, and 3.4).
For slightly asymmetric chromatographic peaks, the assignment of
peak shape models may be ambiguous due to the imprecise measurement of
B/A (e.g., Is a peak with B/A = 1.03 ± 0.02 Gaussian?). Insofar as
accuracy and precision are concerned, does it matter if EMGbased
equations are used with Gaussian peaks or viceversa? As seen from
Table 3.2, the EMG based equations for N , M 2 , and a Q are accurate to
within ±1.5%, ±1.5%, and ± 4%, respectively, over the asymmetry range
1.00 < B/A < 1.09. Figure 3.3 shows the accuracy of the Gaussian based
equations (a = 0.1) over this same range. Clearly, little error in the
estimation of M 2 , N sys , and a Q will result from peak model
misassignments at low asymmetries (1.00 < B/A < 1.09) due to B/A
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imprecision, though for B/A > 1.04 the EMG equations are more accurate.
Furthermore, as seen from Table 3.5, the change in the precision of CFOM
estimation resulting from peak model misassignment would be less
than 1$.
Doublechecking the results . The universality of the relative
error approach (introduced in the Derivations section above) was checked
by the independent variation of t Q and Oq over the t Q / a G range of 10 to
5000. The generality was confirmed by identical statistical results
(£RELs, etc.) over a given B/A range (e.g., 1.092.76) for all CFOMs
calculated from this approach.
As an additional check on the experimental work, M 5 and N_,,_ were
£~ sys
calculated by statistical moment analysis (23), using the same to search
algorithm as before. The agreement among the true, moment, and manual
values for both M 2 and N was within + 1$ for 1.00 < B/A < 2.05 with
very little bias present in either approximation. At higher asymmetries
(2.05 < B/A < 2.76), the manual values remained within + 1.5$ of the
true values, but the corresponding moment values showed a significant
positive bias ranging from +1.5$ to +5.5$. An even greater bias of
+4.4$ at B/A = 2.05 ( T / a G = 2) reported elsewhere (23) was attributed to
arbitrary data truncation. This was probably the source of bias in our
statistical moment method as well, but regardless of the source of bias
the same conclusion may be drawn: at high asymmetries (2.05 < B/A <
2.76) the manual CFOM equation method is more accurate than the moment
method.
Working range of the equations . The EMGbased equations in
Table 3.2 were expected to be accurate over the asymmetry range used for
the least squares curvefitting of the f(B/A) approximations. Thus,
50
except for eq 8, the accurate working range was thought to be 1.09 < B/A
< 2.76. Nevertheless, eqs 1, 2(a,b), 6, 7, and 10 allow accurate
estimation of N , M 2 , t Q , N fflax , and M 1 , respectively, for Gaussian
shaped peaks (B/A = 1.00). Although somewhat surprising, this is
explained by the near convergence of these EMG equations to Gaussian
ones when the substitution B/A = 1.00 is made in the former. The N
sys
equation in Table 3.2, for example, becomes
N sys = l8.53(t R /w 0<1 ) 2 (3.11)
which is within +0.6$ of the Gaussian formula
Vl = 18 ' 42( V W 0.1 )2  (3.12)
CFOM equations could have been developed for asymmetries greater
than B/A = 2.76 (T/a Q = 3), but a sacrifice of simplicity, accuracy, or
both would have been required. More importantly, however, it was felt
that nearly all peaks reported in the literature exhibit B/A's < 2.76.
Indeed, a chromatographic system producing peaks with B/A's > 2.76 is
operating at a relative system efficiency of less than 10%.
When the EMG N s equation was tested for asymmetries higher than
those for which it was developed, the $RE varied between 1.5$ and 10$
for 2.77 < B/A < 4.00, compared to the %RE range of +70$, +110$ for the
Gaussian 10$ equation (N,, ).
w 0. 1
Real versus ideal CFOMs: column characterization . Given that peak
asymmetry is (almost) always present in any real chromatographic system,
N sys' M 2' and fc R ^present the experimentally observed chromatographic
efficiency, peak variance, and retention time, respectively. The
51
corresponding CFOMs N max , a^ , and t Q represent idealized
chromatographic parameters which would describe the system if all
sources of asymmetry could be eliminated. If all or nearly all
asymmetry is extracolumn in origin, then for a given set of conditions
« fflax i a c » and fc G are valid descriptors of the efficiency, band
broadening, and retention characteristics of the column.
Pluralism of the method . As might be surmised from Table 3.2,
there is more than one way to calculate several of the CFOMs. The
variance, for example can be calculated via eq 2a from measurements of
w 0.1 and B/A or via e< 3 2b frora fc R and N S ys ( e 3 1 ) Generally, CFOM
estimates via the "b" equations are simpler, faster, equally precise,
but less accurate than estimates via the "a" equations. This tradeoff
of accuracy for simplicity and speed is slight, however; in most cases
much time can be saved with little sacrifice in accuracy if the "b"
equations are employed.
Only the simplest and most accurate methods are given in Table 3.2.
Therefore, while N could be calculated from its components t R , a Q ,
and t (see N equation in Table 3D, this method was not reported
since it would be much more timeconsuming, tedious, and in all
likelihood less accurate and less precise than the N _ equation in
sys
Table 3.2,
Usefulness of RSE T RPL . The relative system efficiency (RSE) and
relative plate loss (RPL), two new parameters defined in Table 3.1, are
dualistic CFOMs. First, they can be interpreted intuitively as
N sys /N max and (N max " N sys )/N max> respectively, with a corrective
retention factor (t Q /t R ) 2 applied. Alternatively, RSE and RPL can be
viewed as the relative contributions of symmetrical ( a Q 2 ) and
52
asymmetrical (t ) bandbroadening processes to the total system band
broadening (M 2 , the total variance). If (t G /t R ) 2 can be neglected
because of its nearness to unity, the former intuitive expressions for
RSE and RPL become particularly useful. For example, the best possible
efficiency, N max , can be related very simply to the true chromatographic
efficiency, N sys , by
N max = *sys /RSE (3.13)
In fact, this approximation is good to within + 2% with an RSD limit of
+ 5% (for 1.00 < B/A < 2.76) whenever N „ (calculated via eq 3.13) >
4000. Thus, eq 3.13 can serve as a useful estimation of S__„ for
HldA
moderate to high efficiency chromatographic systems.
Although B/A, RSE, and RPL are mutually interdependent (i.e., once
B/A has been measured RSE and RPL may be calculated), the specification
of RSE, RPL, or both in addition to the reporting of B/A greatly
enhances the qualitative description of a chromatographic system. Thus,
while B/A = 1.30 indicates that a given peak is asymmetric, the
corresponding RSE = 55$ or RPL = H5% provides a clearer indication of
the actual efficiency and how much room for improvement exists.
Figure 3.4 shows the exponentiallike relationship of RSE and RPL
with B/A. Chromatographic systems with asymmetries of 1.00 and 1.10 are
operating at much different relative efficiencies, while two systems
operating at B/A = 2.00 and 2.10 are realizing nearly the same relative
efficiencies.
Alternative derivations . Two other approaches for deriving CFOM
equations were not as successful as that already described. In the
first attempt, a modification of the Carr graphical method, three
universal calibration curves were approximated by linear or quadratic
53
RPL
RSE
B/A
Figure 3.4. Relative system efficiency, RSE, and relative plate loss,
RPL, for ideal and skewed peaks.
54
polynomials. This approach yielded equations for a Q , x , t Q , and the
remaining CFOMs in terms of W Q>1> B/A, and t R , but was unsatisfactory
for three reasons: a) an accurate but simple approximation of t/a„ in
u
terms of B/A cannot be obtained for the range 1.00 < B/A < 2.76 because
the relationship between them changes at B/A = 1.36 from a decidedly
nonlinear one to an almost perfectly linear one; b) the errors
introduced by the polynomial approximations tend to accumulate slightly
rather than cancel; and c) the equations derived for all the CFOMs
except x, o G , and t Q are extremely unwieldly, e.g., eq 3.14 below,
N sys = (t fi /w 0>1 ) 2 [ g (B/A)/h(B/A)] (3.14)
where g and h are second degree polynomials of B/A. A variation of this
same approach in which B/A was substituted for the original abscissa
x /°q in the latter two calibration curves made little difference.
A second attempt, a variation of the relative error {RE) approach
utilizing (B  A) as the approximation G in eq 3.5 for t, failed
because the simplest f(B/A) approximation required for sufficient
accuracy was too complex.
CFOM units. For ease of interlaboratory comparison, all non
unitless CFOMs should be reported in time units; if units of length are
chosen instead, the recorder chart speed should be specified to permit
conversion to time units.
55
Conclusion
Superiority of EMGbased Equations . Although Gaussianbased
equations are somewhat simpler than their EMG counterparts in Table 3.2
(compare eq 3.11, text with eq 1, Table 3.2), the EMG equations are
clearly superior because with comparable precision (see Table 3.5) they
are equally accurate for Gaussian (or nearGaussian) peaks and
considerably more accurate for skewed (EMG) chromatographic peaks.
CHAPTER 4
CLARIFICATION OF THE LIMIT OF DETECTION IN CHROMATOGRAPHY
Introduction
In this chapter, the focus is on eliminating the confusion
surrounding the limit of detection (LOD) in chromatography. Some prior
knowledge of the LOD concept is assumed. This discussion primarily
applies to both gas and liquid chromatography using concentration or
mass sensitive detectors whose output is measured as peak height, though
two sections (Eliminating Mistak en Identitiaa and Using the Correct Units 1
apply to analyses using peak area as well. Ideal linear elution, i.e.,
Gaussian peak profiles, is assumed though moderate deviations can be
tolerated,,
Identifying Current Problems
Literature Survey Results
Initially, to determine the sources of discrepancy in
chromatographic detection limits, we conducted a limited survey of
analytical textbooks, chromatographic monographs, and the primary
chromatographic literature. This survey revealed two mistakes of
omission, as well as four major sources of discrepancies. All six are
summarized in Table 4.1. The first two problems are blatant omissions
which discredit the work reported, at least to some degree. However,
they can be eliminated if more attention is given during manuscript
preparation, and thus may be dismissed without further discussion. The
56
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58
remaining four problems (36 in Table 4.1) comprise the major sources of
discrepancy in chromatographic detection limits, and are the topics to
be addressed in this report.
Numerical Example
Before proceeding, however, a numerical example which incorporates
problems 46 (see Table 4.1) will be given. This example will
demonstrate to the reader the magnitude of these problems and will
facilitate later discussion. Though designed with liquid chromatography
in mind, the points made by this example apply equally well to gas
chromatography .
The initial assumptions, experimental conditions, and results of
this example are shown in Table 4.2. To avoid the confusion which it
would certainly have caused, problem #3 of Table 4.1 was not
incorporated into this example. If it had been, the results might have
been even more shocking. Nevertheless, as seen in the bottom row of
Table 4.2, the LODs for the two systems employing identical detectors
differ by three and onehalf orders of magnitude!
Though a detailed explanation of this example is beyond the scope
of the present discussion, the huge discrepancy in the two LODs will be
reconciled after the last three problems in Table 4.2 are solved. This
example should awaken the reader to the seriousness of these problems
and demonstrate why they must be eliminated if the chromatographic LOD
is to be a meaningful figure of merit.
59
Table 4.2. Example of widely differing detection limits
Assumptions
1, Liquid chromatograph with UV absorbance detector
2. beer's Law applies, i.e., A = ebc.
3. Analytical sensitivity, S = eb = 10,000 AU L mol
4. Peak to peak noise, N D _ D = 2 x 10"^ AU
5. Root mean square noise, N = 1/5 N
1
Variables
Experiment
A
Experiment ,B
V inj
5 uL
20 uL
LOD defn
10 HL _/S
PP
3 N /S
 J rms
V M (mL)
2.5
0.5
k
10
3
N (plates)
1000
10,000
Results
LODs reported 8.7 x 10" 6 M
log (L0D A /L0D B ) = 3.5 orders of magnitude difference!
3.0 x 10" 9 M
60
Solving the Problems
Eliminating Mistaken Identities
The limit of detection has unfortunately been confused with three
other concepts—particularly the (minimum) detectability— which are also
used in characterizing chromatographic trace analyses. Table 4.3
includes symbols and definitions for all four of these concepts.
One reason that the LOD is confused with the other concepts in
Table 4.3, particularly the MD, is the redundancy in nomenclature of the
detection limit and the MD, as evidenced by the partial, but
representative list of (apparent) synonyms for these concepts (shown in
Table 4.3) which appear frequently in the literature. In general,
redundant terminology in science only serves to confuse. This is
especially true when the apparent synonyms for different concepts are
themselves quite similar. The use of these apparent synonyms should be
discontinued immediately.
Even if the confusion resulting from the redundant terminology
could be eliminated, the LOD might still be confused with the MD by the
apprentice chromatographer because their definitions, as shown in
Table 4.3 and in eqs 4.1 and 4.2, are so similar in appearance (cf.
meaning, however).
LOD = arbitrary detector signal level/S (4.1)
MD = arbitrary detector signal level/S d (4.2)
£et despite their similarities, the LOD and MD are distinct
concepts, as a closer scrutiny of Table 4.3 shows. The LOD is a general
concept characterizing any overall trace analytical procedure consisting
of one or more steps, whereas the MD is a specific term characterizing
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one step in a chromatographic analysis: detection. For example, the
LOD must, by definition, include the chromatographic dilution of the
analyte, whereas the MD cannot. Furthermore, the LOD is measured
experimentally with a complete chromatographic system (including the
column) under the specific conditions of a given trace analysis; the MD,
in contrast, is measured without a column under conditions that may or
may not correspond to those of the analysis.
The two concepts may also be distinguished mathematically.
Assuming an analytical signal in terms of peak height, the relationship
for a concentration sensitive chromatographic system is (36)
LOD = (2tt) 1/2 [V M (l+k)/N 1/2 ] b MD (4.3)
where V M represents the corrected gas holdup volume in GC and the column
void volume in LC; k is the capacity factor; N is the number of
theoretical plates; and b is a unitless parameter which permits the LOD
and the MD to be defined independently of one another. For a mass
sensitive chromatographic system, the relationship between the LOD and
the MD is (37)
LOD = (2tt) 1/2 [t M (1+k)/N 1/2 ] b MD (4.4)
where t M is the retention time of an unretained solute, corrected for
gas compressibility in GC.
One final difference should be noted: An arbitrary detector signal
level of twice the peaktopeak detector noise has been universally
agreed upon for MD calculations (in practice, at least). No such
consensus exists for the LOD.
63
Choosing a Model
As discussed earlier, the LOD may be defined in terms of an
arbitrary signaltonoise (S/l\l) level. In our survey of the
chromatographic trace analysis literature, we found a multitude of
arbitrary S/N levels used, ranging from 2 to 10. To further complicate
matters, neither the measures of the signal (peak height, peak area,
etc.) nor the noise (peaktopeak, root mean square, etc.) were
specified in many instances.
These inconsistencies and ambiguities are not surprising since (to
our knowledge) no standard model for the LOD has ever been proposed,
much less adopted, by any recognized organization for the field of
x
chromatography! We note specifically the omission of an LOD definition
in chromatography by the American Society for Testing and Materials
(ASTM) and by IUPAC in their respective publications on gas and/or
liquid chromatography nomenclature (3841 ) . The omission by these and
other organizations is also substantiated in two reviews (42,43).
More importantly, however, the above inconsistencies and
ambiguities can be eliminated completely if a clearly stated LOD model
is adopted. Therefore the adoption of, with minor reinterpretation, the
IUPAC model for spectrochemical analysis (44) or a model based on first
order error propagation (10) is proposed. These models are given in
eqs 4.5a and 4.5b, respectively,
LOD = 3s B /S (4.5a)
(The model which the International Union of Pure and Applied
Chemistry (IUPAC) adopted in 1975 (44) was chosen specifically
for spectrochemical analysis. Though the ACS Subcommittee on
Environmental Chemistry reaffirmed this model in 1980 (45),
it did so for the area of environmental chemistry and not
specifically for the field of chromatography.)
64
LOD = 3[s B 2 + Si 2 + (i/S) 2 s s 2 ] 1/2 /S
(4.5b)
where S, i, s s , and s j _ are the analytical sensitivity (slope),
intercept, and their respective standard deviations of the calibration
curve obtained via linear regression; and s B is the standard deviation,
calculated from 20 or more measurements of the blank signal.
The factor of 3 in the numerator of the right hand expressions of
eqs 4.5a and 4.5b gives a practical confidence level of 90$ to 99.7$,
depending on the probability distribution of the blank signal and the
accuracy of s B (10,44,46). Though smaller or larger factors could be
used instead of 3, the resulting confidence levels would be too low or
too high for practical use in most cases. Both the original proponents
of these models (10,44) and others (46) strongly recommend the use of
the factor 3. This author concurs.
Both models are proposed for adoption because it seems preferable
to let the chromatography community judge their respective merits.
Indeed, strong arguments can be made for each. The IUPAC model, on the
one hand, is computationally simpler and has already been employed,
though infrequently, in the chromatographic literature. On the other
hand, the error propagation model is not really all that complicated;
many pocket calculators with linear regression capability can be easily
programmed for the error propagation model. Furthermore, the error
propagation LOD model takes uncertainties of the slope and intercept of
the calibration curve into account, resulting in a more realistic
numerical estimate.
Interpretation . The IUPAC and error propagation LOD models were
developed originally for spectroscopic trace analysis. Nearly all the
65
associated concepts, [e.g., the calibration curve, the sample (analyte +
matrix), etc.] have identical, straightforward interpretations in
chromatography. One aspect, however, does not: the measurement of s D .
a
Intuitively it is clear that the chromatographic baseline is
somehow analogous to the blank signal in spectroscopy. We can now refer
to s B as the standard deviation of the chromatographic baseline (noise).
The measurement of Sg remains unclear, however.
One possible procedure would be to estimate Sg from 20 or more
measurements of only that portion of the baseline observed at the
analyte' s retention time in the absence of the analyte (when a blank
solution is injected). This is directly analogous to the measurement of
the blank signal (at the analytical wavelength) in spectroscopy. Such a
literal procedure would require at least 20 injections of blank solution
(20 blank chromatograms!) and is obviously too impractical:
1. It would be much too time consuming!
2. It may require too much blank solution.
3. Variables which affect retention would require strict control.
4. The retention time of the analyte would need to be known very
precisely.
A much more practical procedure becomes apparent if one remembers
that the standard deviation (root mean square) of a random (periodic)
signal can be closely approximated by the quotient of the range (peak
topeak displacement) and a parameter, p, dependent on the type of
signal, i.e.,
Signal = ran §e/p (4.6a)
For the measurement of s B , it should be noted that the range is
equivalent to the peaktopeak noise of the baseline, N , if the
latter is measured over a sufficiently wide region of the chromatogram
which includes the analyte peak. Additionally, the baseline usually
66
results from a normally distributed, random signal for which p = 5.
Thus in most oases the standard deviation of the chromatographic
baseline, Sg, can be estimated from onefifth of the peaktopeak noise,
i.e. ,
S B = N pP /5 (4 * 6b)
Two additional comments regarding the practical procedure for
measuring Sg should be noted:
1. It is recommended that the "sufficiently wide region of the
chroma togram" be at least as wide as 20 base widths of the analyte peak.
2. If systematic fluctuations in the baseline are present, a value (of
p) less than 5 should be used in eq 4.6b. If, for example, a periodic
triangular baseline (possibly resulting from flow pulsations in the
detector cell due to an insufficiently dampened solvent delivery system)
is observed, then p = 3.5 and s B = N _ /3.5.
Using the Correct Units
A common misconception about chromatographic LODs is that they can
be reported in units of concentration rather than amount. The fallacy
of this assumption will be shown below.
Dimensional analysis . One way of deducing the correct units of the
chromatographic LOD is by dimensional analysis. Referring to the
definition of the MD in Table 4.2, it is clear that if the appropriate
units of concentration and mass flux are used for the MD in eqs 4.3 and
4.4 for the concentration sensitive and mass sensitive detector cases,
respectively, the units for the LOD in eqs 4.3 and 4.4 must be in terms
of an amount (e.g., moles, grams, or some multiple thereof).
Another approach via dimensional analysis is to consider the right
hand expression of eq 4.1. Given the definition for the analytical
sensitivity, it is clear that the units for this term (denominator of
67
eq 4.1) should be the quotient of the units of the measured signal and
the units of the independent variable. Therefore, since the units for
the noise expression (numerator of eq 4.1) are the same as those for the
measured signal, the units for the detection limit should be the same as
the units of the independent variable of the calibration curve. Thus,
to decide which units are correct for the LOD, we need only to identify
the units of the independent variable of the calibration curve, i.e., to
determine whether the chromatographic signal depends on the
concentration or amount of analyte injected.
Equations 4.7 and 4.8 (36,37) below show that the signal (peak
height, hp) is directly proportional to the maximum concentration,
C max det' or ^ he max i mum flux, F max <jet ' °^ ^ e G h romat °gr , aphic peak
flowing through the detector, which in turn are directly proportional to
the amount of analyte injected.
h P  C max,det = ^inj » V2 C**> 1/2 /Vr (4.7)
h P * F max,det = ^inj ""* ^ U2 /H (4.8)
Thus the same conclusion is reached once again: The chromatographic LOD
must be given in units of amount, i.e., in moles, grams, or some
fraction thereof.
Identifying faulty logic . Despite these convincing arguments, some
researchers insist on reporting their chromatographic LODs in
concentration units. Their rationale might be as follows:
1. The amount of analyte injected, Q.± n i' is the product of the
concentration of analyte in the sample, C, ,, and the volume of sample
injected, ^ . .
«inj = C inj V inj <*•»
68
2. From eq 4.9 it is clear that the concentration of analyte injected,
C inj' is direc tly proportional to the amount of analyte injected.
C inj = W V inj (4 ' 10)
3. Therefore the relative LOD (in units of concentration), C,, is
proportional to the true, absolute LOD, q L (in units of amount).
C L = 0L /V inj (*.")
Though reached in a straightforward manner, the conclusion stated
above is nevertheless false. The error in reasoning is best described
as an improper or incomplete analogy. In going from a true expression,
eq 4.10, to a false statement, eq 4.11, C. ■ and q. . were replaced by
mj inj r J
two limiting quantities C L and q L , respectively. No analogous
substitution was made for V in , , however, and therein lies the error.
V inj may vary continuously over < V inj < V inj>max , where V injfflax is
some limiting, maximum injection volume to be discussed momentarily.
Unless V. nj = V inj)fflax , eq 4.11 is false.
A numerical example will help demonstrate the absurdity of eq 4.11.
Suppose the true, absolute LOD (q L ) for analyte X had 'been determined
independently by two scientists using the same LC system to be 1 x 10~ 12
mol. If the scientists had used different injection volumes of 5 uL and
50 uL, according to eq 4.11 the relative LODs (C L 's) for the same
chromatographic system would be 2 x 10" 6 M and 2 x 10" 7 M, respectively.
Clearly eq 4.11 is inappropriate.
The correct expression, eq 4.12 below, is obtained by using
V inj,max in Place of V in j . But this expression is of little value
because of the difficulty in obtaining a consistent, precise estimate of
V.
in j ,max"
C L = «L /V inj,max < 4 ' 12 >
69
Problems with estimating the maximum injection volume . Experimen
tally, Vj_ n j max °an be determined by increasing V. ., until column over
loading or some other adverse phenomenon is observed. This operational
definition of V in  max is unsatisfactory, however, because the injection
volume at which these events occur is too dependent on the experimental
conditions, e.g., the sample matrix, the percent loading of the column.
Alternatively, numerous theoretical expressions are available for
the estimation of V in  max . Though many are overly specific, a few are
completely general. Perhaps the best is one which relates V ■ to
the maximum tolerable degradation in resolution (5, p. 289). Our
extension of this expression is reported below as eq 4.13, and is
derived in Appendix C.
V inj,max = 2 (3) 1/2 (y 2 + 2y) 1/2 V M (1 + k)/N 1/2 (4.13a)
where y = the maximum tolerable loss in resolution due to a finite
injection volume.
Another expression based on the same criterion (maximum tolerable
loss in resolution) but derived from slightly different assumptions (36)
is shown in eq 4.13b
V inj,max = < 2K) * V M (1 + *)/N 1/2 (4.13b)
where K is a parameter characteristic of the method of injection;
practical values of K range from 2 to 3.5 (ideal plug injection).
Comparison of eqs 4.13a and 4.13b reveals one of the problems with
theoretical estimates of V^ . max : due to their extreme dependency on
the criterion selected and/or the assumptions of the derivation,
theoretical estimates of V i . max are extremely inconsistent.
70
Another problem with these and other general expressions is that
V inj max is ni S nl y dependent on the value of some arbitrary parameter,
y in this example (eqs 4.13a and 4.13b). If y = 0.04 (k% loss in
resolution), eq 4.13a becomes
V = V, .( 1 + k)/N 1/2 (4 14)
v mj,max >r ' + K;/w ^ ,l4 '
If y were chosen to be 0.01 or 0.10, then V . . _„„ would be equal to
LI I J j ludX
approximately half or one and onehalf times the value predicted by
eq 4.14. Since universal agreement on the value of y or some other
parameter is unlikely, the theoretical methods for estimating V . . mov
XII J y ul3X
will continue to be imprecise.
Summary . For all the reasons discussed above, chromatographic LODs
should always be reported in units of amount, not concentration.
Converting to Chromatographic Reference Conditions
As seen from eqs 4.3 and 4.4, the LOD depends not only on the
detector characteristics (MD) but also on three chromatographic
parameters [k, N, and V^ (or t M )] which characterize the column and the
solute. In this section a method is proposed for taking the effects of
these parameters into account, thereby solving the final problem
associated with the chromatographic LOD (see Table 4.1). Gaussian
elution profiles are assumed, though slight to moderate deviations can
be tolerated.
B'or a concentration sensitive detector, the relationship between
the LOD and the chromatographic parameters is given by
q L = d V M (1+k)/N 1/2 (4.15)
where terms of eq 4.3 [(2ir) 1/2 , b, MD] have been incorporated into a
71
single proportionality constant, d. Alternatively, since V„ = V.,(1 + k)
R " "M v
and N = (V^c^) 2 , we may write
q L = d V R /N 1/2 (4.16)
or q L = d a y (4.17)
where V R is the retention volume (corrected for compressibility in GC)
and cfy is the bandwidth of the chromatographic peak, in volume units.
Equations 4.154.17 are equivalent, and any one of them may be used
to derive an expression which accounts for the effects of these
chromatographic parameters on the LOD. For simplicity, we use eq 4.17.
Consider a LOD, q L1 , obtained under one set of chromatographic
conditions, i.e., o y = a v1< By analogy with eq 4.17,
Q L 1 = da V1 ( 4  18 )
Likewise, for a LOD, q L2 , obtained under a second set of chromatographic
conditions,
%2 = da V2 (4.19)
Dividing eq 4.19 by eq 4.18 and solving for q L2 yields
^L2 = ^ G V2 /a V1 ] QL1 (4.20a)
which can also be written as
q L2 = CV R2 /V R1 ] tN 1 /N 2 ] 1/2 q L1 (4.20b)
or q L2 . LV M2 /V M1 ] [(k 2 + 1)/(k 1 + 1)] [N^Ng] 172 q L1 (4.20c)
Equations 4.20a4.20c are important for two reasons: First, they
can be used to predict the change in the detection limit when switching
72
from one set of chromatographic conditions to another. Thus they are
useful to the analyst interested in lowering the detection limits via
improvements in the chromatographic (rather than the detection) aspects
of the trace analysis. It should be noted, however, that the detection
limits cannot be lowered infinitely by such improvements. Eventually
the analysis will be optimized to the point where further improvements
in the chromatographic aspects will necessitate a reduction in sample
volume which offsets the chromatographic improvements (36). Of course,
in situations where very little sample is available, this reduction in
sample size while maintaining a constant detection limit will be
helpful.
Second, eqs 4.20a4.20c can be used to compare detection limits
obtained under different chromatographic conditions (bandwidths) .
Consider two such detection limits obtained using different
concentration sensitive detectors. These detection limits can be
compared only after one of the two detection limits is converted from
its present value obtained under a certain chromatographic bandwidth
(set of conditions) to a value corresponding to the bandwidth of the
other LODj, or after both LODs are converted to values corresponding to a
third bandwidth. Note that the latter would require an additional
conversion.
Whichever way is chosen, it should be recognized that the final
bandwidth (set of chromatographic conditions) chosen in the LOD
conversion procedure serves as a reference state and that the initial
bandwidth(s) is(are) by definition experimental states. Thus from
eq 4.20a we may write
<*L,ref = £a v>ref /a Vjexp ] q L ,exp ( 4  21 >
73
where the subscripts "ref" and "exp" refer to reference and
experimental, respectively. Similar equations resulting from the
incorporation of this notation into eqs 4.20b and 4.20c are easily
inferred.
The above derivation can be repeated in an analogous fashion for
the case of mass sensitive detectors. The result is
*L,ref = [a t,ref /a t,exp ] ^L,ex P " .22)
where a t is the bandwidth of the peak in time units.
Given eqs 4.21 and 4.22, the analyst now has a method for comparing
detection limits obtained under different chromatographic conditions
(states). Given two or more LODs measured under different
chromatographic states, the analyst merely converts all of them to
values corresponding to a single, arbitrary, reference state.
Though the choice of the reference state is arbitrary, the
selection of the reference state is nevertheless important. For
example, it would be counterproductive for an analyst to employ a
different chromatographic reference state each time a new group of
experimental LODs (measured under different experimental states) were to
be compared, since LODs in separate groups could not be compared if this
was done. For similar reasons, it would also be counterproductive if
each analyst or group of analysts used different reference states.
On the other hand, the reference states by definition must be
different for concentration sensitive and mass sensitive chromatographic
systems. And in addition, typical chromatographic states (bandwidths)
vary considerably in liquid chromatography (LC), packed column gas
chromatography (PGC), and open tubular gas chromatography (OTGC).
74
The logical compromise which is proposed is a fixed reference state
(bandwidth) to be used exclusively for each of the three chromatographic
areas (LC, PGC, and OTGC) in each of the two detector categories
[concentration sensitive (eq 4.20) or mass sensitive (eq 4.21)], giving
a total of six fixed references states. Furthermore, it is proposed
that the LODs resulting from the conversion to these fixed reference
states be referred to as standardized chromatographic LODs. To
emphasize this, eqs 4.21 and 4.22 may be rewritten as
a L,std = [G V,ref /G V,exp ] a L,exp (4 ' 2 3)
and q L,std = K,ref /a t,expJ ^L,exp ^ 24 )
respectively, where the standardized LODs are indicated by the subscript
"std".
Advantages of the chromatographic reference state concept and the
resulting standardized chromatographic LODs . If the values for the
reference states are chosen judiciously, the resulting standardized LODs
will be superior to experimental (hereafter termed conventional ) LODs in
several ways:
First, standardized LODs generally represent a more realistic
measure of the trace analysis capabilities of a given chromatographic
system, thus permitting an analyst to decide whether or not a particular
application reported in the literature will be feasible with his or her
system. Conventional LODs, on the other hand, may be very misleading.
As evidenced in the literature, many trace analyses have been performed
using chromatographic systems which have only been partially optimized,
if at all. As a result, unusually pessimistic (very high) LODs are
obtained. In contrast, some overly optimistic (very low) LODs have been
75
reported in some instances where the particular chromatographic systems
have been optimized more than the typical system could have been.
Second, the use of standardized LODs permits a fair comparison of
trace analysis chromatographic systems in different areas, e.g., OTGC
vs. LC, or with different types of detectors (mass vs. concentration
sensitive). Such a comparison is not valid if conventional LODs are
used, even within a given area using the same type of detector.
Third, the standardized LOD facilitates the prediction of the value
of the LOD under experimental chromatographic conditions. Rearranging
eqs 4.23 and 4.24, one obtains
%,exp = [a V,exp /a V,ref ] ^L,std ( 4 25)
&nd q L,exp = [G t,exp /a t,refJ ^L,std ^.26)
Assuming q LjStd has been reported, one merely needs to substitute values
for a V,exp (or Vexp) in e 3 4 25 °r *• 26 in order to calculate q r
Fourth, the standardized LOD can serve not only as a figure of
merit for the overall chromatographic analysis, but as a figure of merit
for the detector as well. The need for the minimum detectability (MD)
as a separate parameter (and hence, the need for a detector calibration
curve) would therefore be eliminated, resulting in a considerable
savings of time.
Numerically defining the refere nce states . Values for the proposed
chromatographic reference states (bandwidths) were determined by
assigning values to the component parameters [V [v] (or t M ) , k and N] and
then calculating the reference bandwidths using eq 4.27 or 4.28:
a V,ref = V M,ref (1 + k ref )/(N ref )1/2 ^ 2 7)
76
CT t,ref = ^.ref + k ref )/(N ref )1/2 (4.28)
The reference values for the individual parameters were selected in
accordance with one or more of the following criteria:
1. The value represents an intermediate chromatographic performance
which is easily achieved except under unusual circumstances.
2. The value falls within a range of values reported directly in the
scientific and trade literature.
3. The value falls within a range of values calculated from column
manufacturer specifications.
4. The value falls within a previously recommended range [k only— see
ref 12, p. 67].
The values selected for the chromatographic bandwidth component
parameters are given in Table 4.4. The resulting reference states and
corresponding standardized LOD equations are shown in Table 4.5.
Table 4.5 and the advantages enumerated earlier conveniently
summarize the bulk of the material presented in this section. Only one
additional point needs to be made: nonstandard (experimental)
bandwidths are best measured directly using equations such as
G = V 4 (4.29)
° r a G = W 0<1 /4.292 (4 >30)
where W fa and W Q#1 represent the base width and the width at 10% peak
height, in units of volume or time, whichever is appropriate.
Alternatively, the bandwidths can be calculated from their component
parameters, but this is disadvantageous in two respects. First, it
requires at least two measurements instead of one. Second, different
expressions and/or different interpretations are required for gradient
(temperature or mobile phase) elution conditions.
77
Table 4.4. Proposed values for the component parameters
of the chromatographic reference states
parameter LC PGC OTGC
k ref
N ref 10,000
V M>ref (mL)
^.ref^ 1 "') 1
10,000
62,500
3
2
0.3
1.3
78
Table 4.5. Proposed standardized LOD equations and
the corresponding chromatographic reference states
q L,std = tCT V,ref /a V,exp ] q L,exp concentration
sensitive detector
q L,std = ta t,ref /CT t,exp ] q L,exp mass sensitive
detector
LC
PGC
OTGC
°V,ref (mL)
0.05
0.15
0.04
^.ref^ 111 )
0.05
0.015
0.026
79
The Numerical Example  Revisited
We return to our hypothetical LOD example (Table 4.2 and associated
text) to reconcile the large differences in the reported detection
limits. Recall that since identical chromatographic detectors were
employed, the huge discrepancies were attributed solely to problems 46
of Table 4.1. By eliminating these problems one at a time, obtaining
identical detection limits at the conclusion, this claim will now be
proven.
The reconciliation is summarized in Table 4.6, though readers who
wish to perform the calculations will need to refer to conditions
specified in Table 4.2.
The progress of the LOD reconciliation may be noted by inspecting
either the LOD values themselves in the second and third columns, or
their ratio given in the fourth column, in orders of magnitude. The
degrees to which the given problems are responsible for the initial
discrepancy between the LODs are shown in the fifth column; they are
obtained by subtracting successive values in the adjacent column.
Three steps were performed in the reconciliation. First, the LOD
values reported incorrectly in units of concentration (mol L" 1 ) are
converted to the appropriate units of amount (mol) by multiplying by the
corresponding sample injection volumes (V in j's). Second, these unit
corrected LODs are then converted to values consistent with the IUPAC
model previously discussed (q L = 3 s B /S). For case A, since N _ = 5s R
(as discussed earlier in Choosing a Model ), the LOD must be reduced by a
factor of 50/3. For case B, since N rms = s B , no adjustment is
necessary. Finally, the IUPAC consistent LOD values are converted to
80
Table 4.6. Reconciling the differences in the detection limits
from the numerical example in Table 4.2
Step
LOD,
L0D t
log
\ L0 V
log
0. Initial
1 . Amount
8.7 x 10 6 H 3.0 x 10~ 9 M
44 pmol
2. IUPAC def'n 2.6 pmol
3. q L)Std 150 fmol
60 fmol
60 fmol
150 fmol
3.5
2.9
1.6
0.0
0.6
1.3
1.6
81
standardized LODs, thereby adjusting for differences in the
experimental chromatographic conditions (bandwidths, or states).
As 3een in Table 4.6, the discrepancy between the LODs is reduced
significantly with every successive stage. It should be noted that the
largest source of discrepancy is due to differences in the experimental
chromatographic conditions (bandwidths). This demonstrates the need for
a standardized chromatographic limit of detection. Finally, the LODs in
the bottom row of Table 4.6 are identical, indicating that the
reconciliation has been completely successful.
Conclusion
Meaningful chromatographic detection limits can be obtained only if
careful attention is paid to the application of the principles which
have been discussed:
1. The limit of detection (LOD) should not be confused with the
(analytical) sensitivity (S), the minimum detectability (MD), or the
detector sensitivity (S,).
2. The experimentally observed LODs should be calculated using the IUPAC
and/or the error propagation model(s). The calibration curve should be
constructed from a plot of signal versus amount (not concentration!) of
analyte injected, thus insuring that the resulting LODs will be in units
of amount (not concentration!).
3. If obtained under nonstandard conditions, the detection limits can
be standardized using the equations in Table 4.5. Standardized
chromatographic LODs are superior to their conventional (non
standardized) counterparts for several reasons, in particular because
they permit the (valid) comparison of trace analysis chromatographic
systems in different areas (LC, PGC, and OTGC) and/or with different
types of detectors (mass and concentration sensitive).
CHAPTER 5
SUGGESTIONS FOR FUTURE WORK
The results of this work should facilitate and encourage the study
of a number of related and unrelated topics. A few examples are given
below.
With regard to the standardized chromatographic LOD concept
introduced in Chapter 4, if open tubular liquid chromatograpy ever comes
of age, it will be desirable to extend the LOD concept to this area by
defining two additional reference states (bandwidths) .
The successful generation of the exponentially modified Gaussian
(EMG) function by microcomputer and subsequent measurement of the
associated universal data should encourage others to use this realistic
model in related or unrelated modeling studies. One project which has
already been suggested by Kirkland et al. (23) is the development of a
chromatographic resolution function (equation) which is applicable to
skewed peaks as well as to ideal ones.
The equations recommended in Chapter 3 for the characterization of
skewed (EMG) and Gaussian peaks could be tested on analomous peaks
resulting from column overloading, nonlinear distribution isotherms, and
other sources of analomous peak distortion.
Since the graphical deconvolution of the symmetric (^q 2 ) and
asymmetric (t ) sources of band broadening is now possible via eqs 3 and
4 in Table 3.2, the next step would be the development of methods for
measuring the individual components of ov. and i .
82
83
And finally, the relative error (fi£) approach used so successfully
in the derivation of the equations in Chapter 3 is completely general
and should be considered by other scientists for any semiempirical
modeling in their fields.
APPENDIX A— DISCUSSION AND LISTING OF THE BASIC PROGRAM, EMGU
EMGU, the BASIC program which evaluates the EMG function and
obtains the universal data, is listed below. In addition to the
documentation supplied within the program itself and the flowcharts
given in Fig 2.1, the following should be noted:
1. By letting S = a Q / T and Y = (tt Q )/a G , eq 2.4 can be written as
h EMQ (t) = [AS(2tt) 1/2 ] exp(S 2 /2  SY) /exp(y 2 /2)dy/(2T:) 1/2 ( A .1)
J co
where z = Y  S. (A ?)
2. To the extent allowed by BASIC, the symbolism used in EMGU is
consistent with that in eqs A.1 and A. 2 and elsewhere in the text.
Greek symbols were spelled out partially or entirely.
3. EMGU was developed on an Apple II Plus computer and optimized for
0.1 < t/o g < 4 using t Q = 100, a Q = 5, 0.001 < dt fflin < 1, and 0.001 < a
< 1 . Minor modifications may be required for optimum performance if
other computers or other values for the parameters are used.
4. EMGU was designed to minimize execution time; major reductions are
not likely to be achieved unless a compiler is used.
5. The EMG evaluation subroutine (lines 29903160) may be used
independently of the other routines in EMGU.
84
85
100 REM EMGU EVALUATES THE EXPONENTIALLY MODIFIED GAUSSIAN (EMG)
FUNCTION
110 REM ALSO DETERMINES TR, HP, TA, TB, B/A, W/SIG, AND (TRTG)/SIG
120 B$ = •» ADD LINE»:C$ = » 3150 PRINT T,HEMG":D$ = "TO HELP DEBUG."
130 ZERO = 0:PT5 = .5:WUN = 1:TWO = 2:RTTWOPI = SQR (TWO *
3. 141592654): TEN = 10: K4 = 10000:K5 = 100000: REM COMMON NUMERICAL
CONSTANTS
140 P = ,2316419:B1 = .31938153:B2 =  ,356563782:B3 = 1 . 78 1 477937 : B4
=  1.821 255978 :B5 = 1.330274429: REM CONSTANTS FOR 14 POLYNOMIAL
APPROX.
150 A = 1:TG = 100:SIG = 5: REM EMG PREEXPONENTIAL CONSTANTS
200 INPUT "ALPHA = (.005 < ALPHA < 1) ? »; ALPHA
220 PRINT "INITIAL, FINAL VALUES OF TAU/SIGMA (T/S)»: INPUT "(0 1 <
T/S <= 4.25) ? ";R1,R2
240 INPUT "TAU/SIGMA INCREMENT ? ";RSIZE
260 INPUT "DT MIN = ? ( .001 , .01 , . 1 , 1 S) »;TLR
300 REM BEGIN DISK STORAGE OF EMG DATA (OPTIONAL)
400 FOR R = R1 TO R2 STEP RSIZE
440 T = R: GOSUB 3300:R = T:S = WUN / R
480 GOSUB 1000: REM GO TO MAIN EMG DATA ROUTINE
500 BA = (TB  TR) / (TR  XTA):WS1G = (TB  XTA) / SIG:RR = (TR  TG)
/ SIG
540 BA = INT (BA * K4 + PT5) / INT (K4 + PT5):T = RR: GOSUB 3300 :Rfl
600 PRINT R,BA: PRINT WSIG,RR: PRINT
700 TR = ZERO: HP = ZERO: XTA = ZEROrTB = ZERO
720 T1 = ZER0:T2 = ZERO:DT = ZERO:HTEMP = ZERO
800 NEXT R
830 REM END OPTIONAL DISK STORAGE OF EMG DATA
850 END : REM PROGRAM HAS RUN TO COMPLETION
1000 T = TG: GOSUB 3000:HTEMP = HEMG:DT = WUN
1020 T s T + DT: GOSUB 3000
1040 IF HEMG < HTEMP THEN GOSUB 3300: GOSUB 3350: GOTO 1100
1060 HTEMP = HEMG: GOTO 1020
1080 GOSUB 3350
1100 IF DT < = TLR THEN 1160
1120 GOSUB 3450: GOSUB 1620
1140 GOTO 1100
1160 TR = T2  DT:T = TR: GOSUB 3000: HP = HEMG:HLOOK = HP * ALPHA
1200 T = INT (TR + WUN + PT5)
1220 T = T  WUN: GOSUB 3000
1240 IF HEMG > HLOOK THEN 1220
1260 DT = WUN: GOSUB 3400
1280 IF DT < = TLR THEN 1340
1300 GOSUB 3450: GOSUB 1830
1320 GOTO 1280
1340 GOSUB 3500: XTA = T: GOSUB 3000
86
1400 T = INT (TR + PT5)  INT (SIG + PT5)
1420 T = T + INT (SIG + PT5): GOSUB 3000
1440 IF HEMG > HLOOK THEN 1420
1460 TUT INT (SIG + PT5):T2 = T:DT = WUN: GOSUB 2030
1460 IF DT < = TLfi THEN 1540
1500 GOSUB 3450: GOSUB 2030
1520 GOTO 1460
1540 GOSUB 3500.TB = T: GOSUB 3000
1560 RETURN : REM END OF MAIN EMG DATA ROUTINE
1600 REM TR SEARCH LOOP BELOW THRU LINE 1720
1620 FOR T = T1 TO T2 + DT STEP DT
1640 GOSUB 3000
1660 IF HEMG < HTEMP THEN GOSUB 3300: GOSUB 3350: RETURN
1660 HTEMP = HEMG
1700 NEXT T
1720 A$ = "THE TR SEARCH HAS FAILED!": GOSUB 3550: END
1800 REM TA SEARCH LOOP BELOW THRU LINE 1950
1830 FOR T = T2 TO T1  DT STEP  DT
1860 GOSUB 3000
1890 IF HEMG < HLOOK THEN GOSUB 3300: GOSUB 3400: RETURN
1920 NEXT T
1950 A$ = "THE TA SEARCH HAS FAILED!": GOSUB 3550: END
2000 REM TB SEARCH LOOP BELOW THRU LINE 2150
2030 FOR T = T1 TO T2 + DT STEP DT
2060 GOSUB 3000
2090 IF HEMG < HLOOK THEN GOSUB 3300 :T = T  DT: GOSUB 3400: RETURN
2120 NEXT T
2150 A$ = "THE TB SEARCH HAS FAILED!": GOSUB 3550: END
2990 REM EMG EVALUATION SUBROUTINE BELOW THRU LINE 31 60
3000 I = (T  TG) / SIG:E = S * S / TWO  S * Y:Z = Y  SiZTEMP = ZERO
3020 IF Z > = ZERO THEN 3060
3040 ZTEMP = Z:Z = ABS (Z)
3060 NF = EXP (  Z * Z / TWO) / RTTWOPI:Q = WUN / (WUN + P * Z)
3080 PQ = * (B1 + Q * (B2 + Q * (B3 + Q * B4 + B5 * Q * Q)))
3100 I = NF * PQ
3120 IF ZTEMP > = ZERO THEN I = WUN  I
3140 HEMG = A * S * RTTWOPI * EXP (E) * I
3160 RETURN
3300 T = INT (T * K5 + PT5) / INT (K5 + PT5): RETURN
3350 T1 = T  TWO * DT:T2 = T: HTEMP = ZERO: RETURN
3400 T1 = T:T2 = T + DT: RETURN
3450 DT = DT / TEN: RETURN
3500 T = (T1 + T2) / TWO: GOSUB 3300: RETURN
3550 PRINT CHR$ (7): PRINT A$;B$: PRINT C$: PRINT D$: RETURN
APPENDIX B— UNIVERSAL EMG DATA AT a = 0.05, 0.10, 0.30, 0.50
T/a G
0.3
0.35
0.4
0.45
0.5
0.55
0.6
0.65
0.7
0.75
0.8
0.85
0.9
0.95
1.0
1.05
1.1
1.15
1.2
1.25
1.3
1.35
1.4
1.45
1.5
1.55
1.6
1.65
1.7
1.75
1.8
1.85
1.9
1.95
2.0
0.05
(B/A >0.05
W 0.05 /CT G
(t R t Q )/a G
1.0331
5.1088
0.2792
1.0490
5.1812
0.3194
1.0679
5.2620
0.3576
1.0898
5.3506
0.3938
1.1141
5.4468
0.4280
1.1411
5.5494
0.4608
1.1701
5.6582
0.4920
1.2009
5.7722
0.5218
1.2337
5.8912
0.5502
1.2679
6.0146
0.5774
1.3037
6.1416
0.6034
1.3404
6.2722
0.6284
1.3783
6.4056
0.6522
1.4170
6.5414
0.6752
1.4563
6.6796
0.6974
1.4964
6.8196
0.7186
1.5368
6.9610
0.7392
1.5776
7.1040
0.7590
1.6189
7.2482
0.7782
1.6605
7.3934
0.7966
1.7022
7.5392
0.8146
1.7441
7.6862
0.8320
1.7862
7.8336
0.8488
1.8285
7.9816
0.8650
1.8706
8.1302
0.8610
1.9132
8.2792
0.8964
1.9555
8.4286
0.9114
1.9982
8.5784
0.9258
2.0407
6.7286
0.9400
2.0833
8.8788
0.9538
2.1259
9.0296
0.9672
2.1685
9.1804
0.9604
2.2111
9.3314
0.9932
2.2535
9.4826
1.0058
2.2961
9.6338
1.0180
87
T/a G ""^0.05 W 0.05 /a G (t R t )/a G
2  05 2.3367 9.7856 1.0298
21 2.3813 9.9370 1.0414
2  1 5 2.4237 10.0890 1.0528
2.2
2.25
2.3
2.35
2.4
2.45
2.5
2.65
2.75
2.8
2.85
2.95
(B/A) 0>05
W 0.05 /a G
2.3387
9.7856
2.3813
9.9370
2.4237
10.0890
2.4662
10.2406
2.5084
10.3926
2.5509
10.5448
2.5934
10.6968
2.6355
10.8490
2.6778
11.0010
2.7200
11.1534
2.7620
11.3056
2.8044
11.4580
2.8465
11.6104
2.8885
11.7628
2.9303
11.9152
2.9723
12.0678
3.0143
12.2202
3.0561
12.3726
3.0978
12.5252
3.1395
12.6776
1.0640
1.0750
1.0856
1.0960
1.1064
1.1164
1.1262
2 ' 55 2.7620 11.3056 1.1360
2.6
1.1454
1.1546
27 2.8885 11.7628 1.1638
1.1728
1.1816
1.1902
2 9 3.0561 12.3726 1.1988
1.2072
30 3.1395 12.6776 1.2154
89
Va Q ( B /A) 0<1
0.1 1.0013
0.15 1.0043
0.2 1.0096
0.25 1.0174
0.3 1.0278
0.35 1.0406
0.4 1.05&0
0.45 1.0734
0.5 1.0927
0.55 1.1140
0.6 1.1367
0.65 1.1608
0.7 1.1864
0.75 1.2133
0.8 1.2412
0.85 1.2700
0.9 1.3001
0.95 1.3309
1.0 1.3621
1.05 1.3942
1.1 1.4269
1.15 1.4599
1.2 1.4935
1.25 1.5274
1.3 1.5615
1.35 1.5960
1.4 1.6307
1.45 1.6658
1.5 1.7006
1.55 1.7358
1.6 1.7712
1.65 1.8066
1.7 1.8419
1.75 1.8775
1.8 1.9128
1.85 1.9466
1.9 1.9843
1.95 2.0198
2.0 2.0555
= 0.10
W 0.1 /G G
(t R t G )/a Q
4.3142
0.0940
4.3412
0.1440
4.3764
0.1916
4.4200
0.2366
4.4710
0.2792
4.5292
0.3194
4.5932
0.3576
4.6628
0.3938
4.7372
0.4282
4.8158
0.4608
4.8984
0.4920
4.9844
0.5218
5.0736
0.5502
5.1660
0.5774
5.2608
0.6034
5.3584
0.6284
5.4580
0.6522
5.5598
0.6752
5.6632
0.6974
5.7684
0.7186
5.8748
0.7392
5.9S28
0.7590
6.0918
0.7782
6.2016
0.7966
6.3124
0.8146
6.4242
0.8320
6.5364
0.8486
6.6492
0.8659
6.7626
0.8810
6.8764
0.8964
6.9908
0.9112
7.1054
0.9258
7.2204
0.9400
7.3356
0.9538
7.4512
0.9672
7.5670
0.9804
7.6830
0.9932
7.7992
1.0058
7.9154
1.0180
90
Va G
2.05
2.1
2.15
2.2
2.25
2.3
2.35
2.4
2.45
2.5
2.55
2.6
2.65
2.7
2.75
2.6
2.85
2.9
2.95
3.0
;b/aj 0#1
W 0.1 /a G
(t fi t G )/a G
2.0913
8.0320
1.0298
2.1270
8.1486
1.0414
2.1627
8.2652
1.0528
2.1985
8.3822
1.0640
2.2340
8.4992
1.0750
2.2697
8.6160
1.0856
2.3054
8.7332
1.0960
2.3409
8.8504
1.1064
2.3765
8.9676
1.1160
2.4122
9.0850
1.1262
2.4475
9.2024
1.1360
2.4831
9.3198
1.1454
2.5184
9.4374
1.1548
2.5541
9.5546
1.1638
2.5893
9.6722
1.1728
2.6249
9.7898
1.1816
2.6603
9.9074
1.1902
2.6955
10.0248
1.1988
2.7307
10.1426
1.2072
2.7659
10.2602
1.2154
91
0.30
x/a G
0.3
0.35
0.4
0.45
0.5
0.55
0.6
0.65
0.7
0.75
0.8
0.85
0.9
0.95
1.0
1.05
1.1
1.15
1.2
1.25
1.3
1.35
1.4
1.45
1.5
1.55
1.6
1.65
1.7
1.75
1.8
1.85
1.9
1.95
2.0
(B/A)
0.3
1.0185
1.026b
1.0366
1.0474
1.0592
1.0722
1.0857
1.0998
,1149
,1303
,1463
1627
1800
1973
1.2151
1.2334
1.2520
1.2709
1.2900
1.3095
1.3293
1.3493
1.3696
1.3902
1.4108
1.4317
1.4527
1.4742
1.4955
1.5170
1.5387
1.5606
1.5825
1.6043
1.6263
W 0.3 /a G
(t R t G )/a G
3.2248
0.2792
3.2616
0.3194
3.3014
0.3576
3.3434
0.3938
3.3878
0.4282
3.4336
0.4608
3.4806
0.4920
3.5290
0.5218
3.5784
0.5502
3.6288
0.5774
3.6800
0.6034
3.7316
0.6284
3.7844
0.6522
3.8374
0.6752
3.8914
0.6974
3.9456
0.7186
4.0004
0.7392
4.0558
0.7590
4.1114
0.7782
4.1678
0.7966
4.2244
0.8146
4.2814
0.8320
4.3388
0.8488
4.3966
0.8650
4.4546
0.8810
4.5132
0.8964
4.5718
0.9114
4.6308
0.9258
4.6900
0.9400
4.7496
0.9538
4.8094
0.9672
4.8692
0.9804
4.9294
0.9932
4.9898
1.0058
5.0504
1.0180
92
T/a Q
2.05
2.1
2.15
2.2
2.25
2.3
2.35
2.4
2.45
2.5
2.55
2.6
2.65
2.7
2.75
2.8
2.85
2.9
2.95
3.0
(B/A) 03
W 0.3 /a G
(t fi t G )/a G
1.6486
5.1108
1.0298
1.6710
5.1716
1.0414
1.6933
5.2326
1.0528
1.7155
5.2936
1.0640
1.7379
5.3548
1.0750
1.7604
5.4160
1.0850
1.7829
5.4774
1.0960
1.8053
5.5388
1.1064
1.8279
5.6004
1.1164
1.8507
5.6620
1.1262
1.8730
5.7236
1.1360
1.8959
5.7854
1.1454
1.9186
5.8472
1.1548
1.9413
5.9090
1.1638
1.9639
5.9710
1.1728
1.9865
6.0328
1.1816
2.0092
6.0948
1.1902
2.0317
6.1568
1.1988
2.0544
6.2188
1.2072
2.0771
6.2810
1.2154
93
t/c g (B/A) 0<e
0.3 1.0133
0.35 1.0195
0.4 1.0264
0.45 1.0341
0.5 1.0425
0.55 1.0515
0.6 1.0606
0.65 1.0705
0.7 1.0807
0.75 1.0911
0.8 1.1018
0.85 1.1127
0.9 1.1242
0.95 1.1354
1.0 1.1471
1.05 1.1591
1.1 1.1709
1.15 1.1832
1.2 1.1954
1.25 1.2079
1.3 1.2204
1.35 1.2331
1.4 1.2460
1.45 1.2593
1.5 1.2720
1.55 1.2852
1.6 1.2984
1.65 1.3120
1.7 1.3254
1.75 1.3390
1.8 1.3527
1.85 1.3663
1.9 1.3802
1.95 1.3940
2.0 1.4079
0.50
W 0.5 /CT G
(t R t Q )/a G
2.4442
0.2792
2.4706
0.3194
2.4986
0.3576
2.5284
0.3938
2.5588
0.4282
2.5902
0.4608
2.6226
0.4920
2.6552
0.5218
2.6882
0.5502
2.7214
0.5774
2.7550
0.6034
2.7888
0.6284
2.8226
0.6522
2.8568
0.6752
2.8908
0.6974
2.9252
0.7186
2.9594
0.7392
2.9940
0.7590
3.0284
0.7782
3.0628
0.7966
3.0974
0.8146
3.1322
0.8320
3.1668
0.8488
3.2014
0.8650
3.2362
0.8810
3.2710
0.8964
3.3060
0.9114
3.3408
0.9258
3.3756
0.9400
3.4108
0.9538
3.4458
0.9672
3.4808
0.9804
3.5160
0.9932
3.5512
1.0058
3.5864
1.0180
94
Va G
2.05
2.1
2.15
2.2
2.25
2.3
2.35
2.4
2.45
2.5
2.55
2.6
2.65
2.7
2.75
2.8
2.85
2.9
2.95
3.0
(B/A)
0.5
1.4222
1.4364
1.4505
1.4646
1.4789
1.4933
1.5078
1.5220
1.5366
1.5511
1.5656
1.5804
1.5953
1.6098
1.6245
1.6393
1.6543
1.6690
1.6837
1.6986
W 0.5 /CT G
(t R t Q )/a G
3.6216
1.0298
3.6570
1.0414
3.6924
1.0528
3.7280
1.0640
3.7634
1.0750
3.7988
1.0856
3.8344
1.0960
3.8702
1.1064
3.9056
1.1164
3.9414
1.1262
3.9772
1.1360
4.0130
1.1454
4.0486
1.1546
4.0844
1.1638
4.1204
1.1728
4.1564
1.1816
4.1922
1.1902
4.2282
1.1988
4.2644
1.2072
4.3002
1.2154
APPENDIX C DERIVATION OF V .
in j ,max
We begin with a previously derived expression (5, p. 289) which
relates the peak volume observed for a finite size sample, V , to the
volume of injected sample, V" in , , and to the peak volume observed for a
very small sample, V
V = V p 2 +^3 (V inj ) 2 (A.1)
Let V w = (y + 1)V p (A. 2)
where y = loss in resolution due to a finite injection volume.
Substituting eq A. 2 into eq A.1 and solving for V. . yields
v inj = C3/My 2 + 2y)] 1/2 v p (a. 3)
We now express V p in terms of N, k, and V M using eqs A.4A.6, assuming
V p = V b' the volurae corresponding to the base width of the peak.
N = ( V fi/ a V ) 2 (A. 4)
V b = ^ a y (A5)
V R = V M (1 + k) (A. 6)
The result is V p = 4V M (1 + k)/N 1/2 (A. 7)
Substitution of eq A. 7 into eq A. 3 yields the desired expression
V inj = C3/4(y 2 + 2y] 1/2 4V M (1 + k)/N 1/2 (A. 8)
If y is redesignated to be the maximum tolerable loss in resolution,
then eq A. 8 becomes, upon rearrangement,
V inj,max = 2 ^) 1/2 (y 2 + 2y) 1/2 v M (1 + k)/N 1 / 2 ( A . 9 )
which is the desired expression.
95
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BIOGRAPHICAL SKETCH
Joe Preston Foley was born in Lexington, Kentucky on October 23,
1956. He grew up on a livestock and tobacco farm outside Versailles,
Kentucky, and attended the Woodford County public schools. His early
adolescent interests included camping, sandlot football and basketball,
comic books, fireworks, model rockets, and space exploration. He added
running to his list of hobbies one day after he woke his sister (who had
fallen asleep while sunbathing) with a firecracker.
While attending junior high school, Joe was elected President of
the student council and Governor of the statewide Junior Kentucky Youth
Assembly. As he entered Woodford County High School, his extra
curricular interests shifted to chess and varsity tennis. In June of
1974, he graduated with highest distinction (coValedictorian).
During the next four years Joe attended Centre College of Kentucky,
where he was inducted into Phi Beta Kappa. He received chemistry awards
sponsored by Centre as a freshman, sophoaore, and senior. During the
summer of 1977 he worked as a student intern for the Center for
Technology Assessment and Policy Studies (CTAPS) at RoseHulraan
Institute of Technology in Terre Haute, Indiana. In June of 1978, he
graduated with highest distinction (coValedictorian) and received a
Bachelor of Science in chemistry and chemical physics.
In the summer of 1978 Joe again worked for the Center for Technology
Assessment and Policy Studies. In the fall he returned to Centre
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College, his alma mater, for a ninemonth appointment as a physical
science laboratory instructor.
Joe began his graduate study in chemistry at the University of
Florida in June of 1979. In May of 1963 he was one of eleven graduate
students selected nationally for an American Chemical Society Division
of Analytical Chemistry Fellowship. Later in 1983 he completed the
requirements for the degree of Doctor of Philosophy in analytical
chemistry and accepted a National Research Council Postdoctoral Research
Associateship at the National Bureau of Standards in Gaithersburg,
Maryland .